Modal Properties of the Macaw Palm Fruit-Rachilla System: Characterization by Vibration Tests and Numerical Simulations

Abstract

The macaw palm is a palm tree with wide geographic distribution. The aim of this work was to determine the modal properties of the macaw palm fruit-rachilla system subjected to mechanical vibrations. Three-dimensional models were elaborated using a computational program for 3D CAD modeling. The three-dimensional model developed was validated as a function of the real displacements observed during laboratory vibration tests. The average values observed for the system natural frequencies in the vibration tests were 20.09 Hz for the immature stage and 19.86 Hz for the mature stage. The average values ​​of natural frequencies obtained by the numerical simulations were 28.41 Hz and 25.56 Hz for the immature and mature maturation stages, respectively. The vibration mode corresponding to the natural frequencies evaluated was characterized as pendulum displacements of the macaw palm fruit-rachilla system.

Keywords:
finite element method; palm; natural frequencies; vibration modes

HIGHLIGHTS

FEM applications in agricultural mechanization.

Macaw palm harvest.

Modal behavior of macaw palm.

INTRODUCTION

Macaw palm (Acrocomia aculeata (Jacq.) Lodd.) is a palm tree of wide geographic distribution. In Brazil, its main incidence occurs in the Cerrado and Pantanal, with approximately 11.5 million hectares of native culture [¹]. Its main use is based on oil extraction from mesocarp and nut, with numerous applications in industrial and energy sectors such as: biodiesel production, hygiene and cleaning products, cosmetics and food products [²]. Compared to other crops, the estimated oil yield of macaw palm is too high, 4,500 liters per hectare [³].

Currently, the macaw palms exploration is extractivist and the time of harvest is determined when the fruits begin to detach by themselves from the bunches. The fruits can be harvested directly on the trees using scythes to cut the bunches or can be collected on the ground [⁴].

Studies about the macaw palm culture have been developed in the sense of knowing its physical and chemical characteristics [⁵-⁹], in order to infer about the quality of the oil, almond and other parts of the fruits, these studies will allow the controlled and large scale production. Machines that use the mechanical vibrations principle for harvesting have been applied to several crops, such as coffee, olives and pine cult [¹⁰-¹²]. This principle can be considered as a viable alternative to be applied on the detachment of macaw palm fruits, improving harvesting efficiency and the quality of the fruits harvested [¹³, ¹⁴].

The analysis of a vibratory system can be done from computational modeling, in which its input and output variables depend on time and, considering its complexity, it is necessary to simplify the models using only the main characteristics. Several methods can be used for solving differential equations that govern the dynamic behavior of mechanical systems, highlighting analytical and numerical methods. The solution of equations of motion for mechanical systems will provide information about displacements, velocity and acceleration [¹⁵].

Among the numerical methods employed on the analysis of dynamic systems, finite element method is commonly used to address multi-physical problems by researchers and engineers. This numerical technique allows the solution of computational models of physical systems through simulations with a high degree of reliability and can be applied on several areas of engineering [¹⁵-¹⁷], including modeling of the dynamic behavior of agricultural products, such as coffee [¹⁰, ¹⁸, ¹⁹], and having already been used even for macaw palm [²⁰].

Modal and numerical analysis is a method used to determine the modal properties of mechanical systems, corresponding to natural frequencies and vibration modes [²¹]. In a modal analysis it is possible to understand how a structure vibrates, to correlate and update simulation models and to accelerate structural and maintenance calculations. Its validation can be carried out based on the difference between the response of the computational models and experimental data, which allow obtaining models for study of the mechanical systems [¹⁵].

Considering the hypothesis that the principle of mechanical vibrations can be used in the mechanized harvesting process of macaw palm fruits, the aim of this work was to determine the modal properties deterministically, natural frequencies and vibration modes of the macaw palm fruit-rachilla system and validate these models using vibration tests, considering different maturation stages and plants from different sites (accessions).

MATERIAL AND METHODS

The research was carried out using macaw palm bunches collected from the following accessions: Abaeté, MG (BD27); Pitangui/ Martinho Campos, MG (BD40); Prudente de Moraes/ Matozinhos, MG (BGP29) and Mirandópolis, SP (BGP35), in a germplasm bank.

For the determination of the physical properties, computational simulations and vibrations tests were considered two stages of maturation (immature and mature), four accessions (BD27, BD40, BGP29, BGP35) and ten replicates for each accession.

The geometrical, physical and mechanical properties studied in this work were determined experimentally by Vellosoand coauthors [⁶].

Bunches were always collected in the morning and the tests were carried out within 24 hours after collection. Bunches used for the tests were randomly selected in palm trees, which were also randomly selected. As collection took place in a germplasm bank dedicated to research in different areas, the number of samples was reduced, making it not possible to conduct research outside of a completely randomized scenario. The samples used for the vibration tests were composed by one rachilla with one fruit, fruit-rachilla systems were cut directly from the bunches (Figure 1).

Figure 1
Fruit-rachilla samples of macaw palm used in vibration tests.

In vibration tests for bunches with immature fruits, were used samples with a length of 15 centimeters for the accessions BD27 and BGP35, and 13 centimeters for the accessions BD40 and BGP29, whereas in vibration tests for bunches with mature fruits the lengths were 12.0centimeters for BD40 accession; 13.0 centimeters for BD27 and BGP29 accessions, and 15.0 centimeters for BGP35 accession. This difference of lengths is due to the heterogeneity of the bunches' development. The samples were fixed using 2.0 centimeters of the rachilla at the opposite extremity of the fruits (as a cantilever structure) and the accelerometers were fixed directly to the fruits in the direction of vibration (Figure 2).

Figure 2
Rachilla sample with one fruit (A) crimped at one of the ends, in a vibration test using a LDS M6-CE V555 electromagnetic vibrator (B) and a PCB Piezotronics 352C33 high-sensitivity accelerometer to measure the sample (C1) and control the test (C2).

The accelerometers used were uniaxial, for this reason the samples evaluated were composed of only one fruit. From samples with more fruits there could be an influence on the direction of vibration, besides creating an extra weight in the rachilla, resulting in lack of accuracy of the acquired data during the tests performed. Each accelerometer used had a mass less than ten percent of the total mass of the system analyzed, in order to minimize its interference in the dynamic response of the system [²²].

Computational modeling of the fruit-rachilla system

Physical properties (elasticity modulus, Poisson’s ratio and specific mass) were used as input parameters for computational simulation of the dynamic behavior of the fruit-rachilla system. The values used for elasticity modulus were the means of each accession, at each maturation stage (Table 1).

Table 1
Average values for elasticity modulus for each evaluated accession and different stages of maturation

Elasticity modulus (MPa)
Stage	BD27	BD40	BGP29	BGP35
Immature	307.84	304.61	337.56	196.93
Mature	184.12	305.59	247.54	121.88

• Source: mechanical properties obtained by Velloso and coauthors [⁶].

Averages Poisson’s ratios of 0.34 and 0.36 were used for the immature and mature stages of maturation, respectively [⁶].

Specific mass of each structure that composes the fruit-rachilla system was determined from the ratio between the mass and the volume, considering the load of the accelerometer coupled to the system. A precision digital balance, with a 0.001 g of accuracy, was used to determine the mass of each structure (fruit and rachilla). The volume was determined by means of Archimedes immersion method from a graduated cylinder with a minimum graduation of 0.1 mL, in which the fruits and the rachillas were immersed in water. Different values of the specific mass for the fruit and for the rachilla were used for each sample, within each accession for each stage of maturation (Tables 2 and 3).

Table 2
Values for specific mass for each fruits sample, within each evaluated accession and different stages of maturation

Specific mass of fruits (g.cm-3)
	Immature				Mature
Samples	BD27	BD40	BGP29	BGP35	BD27	BD40	BGP29	BGP35
1	1.43	1.41	1.22	0.78	1.42	1.39	1.38	1.44
2	1.40	1.44	1.22	1.27	1.48	1.31	1.45	1.39
3	1.42	1.39	1.23	1.32	1.56	1.49	1.37	1.28
4	1.39	1.33	1.24	1.27	1.44	1.41	1.40	1.32
5	1.35	1.33	1.20	1.27	1.37	1.50	1.30	1.45
6	1.34	1.36	1.24	1.37	1.38	1.20	1.48	1.52
7	1.37	1.35	1.24	1.34	1.36	1.23	1.36	1.41
8	1.42	1.79	1.26	1.32	1.65	1.46	1.43	1.37
9	1.43	1.35	1.19	1.34	1.65	1.53	1.41	1.40
10	1.38	1.43	1.29	1.38	1.49	1.53	1.44	1.49

Table 3
Values for specific mass for each rachillas sample, within each evaluated accession and different stages of maturation

Specific mass of rachillas (g.cm-3)
	Immature				Mature
Samples	BD27	BD40	BGP29	BGP35	BD27	BD40	BGP29	BGP35
1	0.45	0.49	0.52	0.60	0.43	0.51	0.67	0.83
2	0.57	0.40	0.50	0.74	0.65	0.52	0.55	0.71
3	0.49	0.40	0.63	0.78	0.63	0.47	0.59	0.67
4	0.55	0.39	0.44	0.55	0.53	0.46	0.69	0.76
5	0.63	0.41	0.55	0.71	0.48	0.70	0.71	0.83
6	0.75	0.38	0.64	0.62	0.48	0.46	0.52	0.72
7	0.49	0.35	0.57	0.69	0.60	0.36	0.79	0.72
8	0.42	0.40	0.57	0.69	0.53	0.28	0.83	1.00
9	0.56	0.34	0.64	0.75	0.48	0.41	0.80	0.67
10	0.53	0.53	0.52	0.62	0.44	0.38	0.70	0.83

The three-dimensional models were elaborated from the 3D CAD software, in which the geometric characteristics were used to develop the virtual models of the systems respecting their real forms. A spheroid with three distinct diameters was used to represent the fruit that was attached to a quarter of the rachilla extremity (Figure 3). The connection between fruit and rachilla was made by a small conical structure, and it was considered rigid during the simulations. In each simulation model were used the average geometric characteristics (rachillas lengths and, rachillas and fruits diameters) corresponding to the fruit-rachilla system submitted to the vibration tests. For all simulations performed, the boundary condition applied to fruit-rachilla systems was constraint displacements in x, y and z directions at the end of the rachilla.

Figure 3
Macaw palm fruit-rachilla system model.

Autodesk Fusion 360 software was employed to perform the finite element analysis and to determine and simulate the dynamic behavior of the system. This software allowed the discretization of the systems (from the generation of unstructured meshes for three-dimensional domains, using small volumes which consist in elements and nodes), mathematical modeling and solution, as well as the analysis of numerical results (from graphs, diagrams and animations).

For the discretization of the models were used a quadratic approximation function and ten-node parabolic tetrahedral elements with a mesh with 14,773 nodes and 9,098 elements, which allow the elements to represent the edges better with a significant computational cost reduction. The models were considered linear, elastic and isotropic for the simulations.

Therefore, from the models it was possible to study the dynamic response of the fruit-rachilla systems of the macaw palm submitted to mechanical vibrations, considering different scenarios.

Determination of natural frequencies and vibration modes

Modal properties of the fruit-rachilla system were obtained based on the models developed. The natural frequencies of the fruit-rachilla system (eigenvalues), as well as its respective vibration modes (eigenvectors) were obtained from the formulation and solution of the eigenvalue and eigenvector problems.

In order tomodel it, damped free vibration systems with multiple degrees of freedom were employed, represented by differential equations, which can be expressed in matrix form (Eq. 1).

$\left[M\right].\left\{\ddot{\nu }\right\}+\left[K\right].\left\{\nu \right\}=\left\{0\right\}$ (1)

where,

[M] = mass matrix, kg;

{$\ddot{\nu }$} = acceleration vector, m/s²;

[K] = stiffness matrix, N/m;

{ν} = displacement vector, m.

The determination of the eigenvalues was obtained assuming that the undamped free vibrations are harmonic (Eq. 2).

$\left\{v\right\}=\left\{{\phi }_i\right\}.\cos \left({\omega }_i.t\right)$ (2)

where,

{φ_i} = represents the eigenvector associated with i-th natural frequency of the system;

ω_i = angular frequency, rad/s;

t = time, s.

The derivative of Eq. (2) with respect to time and its substitution in Eq. (1), results in Eq. (3), which allows the determination of natural frequencies (eigenvalues) and vibration modes (eigenvectors).

$\left(-{\omega }^2.\left[M\right]+\left[K\right]\right).\left\{{\phi }_i\right\}=\left\{0\right\}$ (3)

Thus, making the determinant of the matrix $(-{\omega }^2.\left[M\right]+\left[K\right])$equals to zero, the natural frequencies and the mode shapes of the system can be obtained [¹²,¹⁶,²³,²⁴].

The calculation of the eigenvalues and eigenvectors was performed by Lanczos numerical method, which is used in a solution of several engineering problems of solid mechanics, allowing simulations with a fast convergence rate and a low spend of memory.

Validation of model in finite element method

The three-dimensional model was validated using the actual displacements obtained from vibrations tests. The vibration of samples were performed by an electromagnetic shaker, model V555 M6-CE manufactured by LDS (Ling Dynamic Systems), associated with a vibration controller model COMET_USB and a power amplifier model PA 1000L connected to a field power supply model FPS (Figure 4).

Figure 4
Measuring system of vibration response.

In vibrations tests, the samples were subjected to a frequency sweep in the range of 10 to 40 Hz, considering 1.5 octaves per minute and peak-to-peak displacement of 1.0 mm during 90 seconds. This setup enabled the samples of the fruit-rachilla system to pass through resonant frequencies. Acceleration amplitudes were measured by high sensitivity accelerometers (100.7mv/g (Eu)), manufactured by PCB piezotronics, model 352C33.

For acceleration data acquisition, a code in LabView software was developed [²⁵]. This code managed a data acquisition system made by National Instruments, model NI cDAQ-9174, with NI9234 four-channel module for acceleration acquisition signal, using a sampling range of 400 Hz.

The acquired acceleration data in the time domain were transformed to frequency domain using the Fast Fourier Transform (FFT). Therefore, based on frequency spectra, it was possible to determine the experimental natural frequencies and compare them to the frequencies obtained from numerical simulations.

The analysis of the difference between experimental and simulated natural frequencies was carried out by Tukey test at 5% of probability.

Statistical analyses were performed using the R statistical program [²⁶].

RESULTS

Experimental natural frequencies were identified in frequency spectra from the amplification of the amplitude of vibration at a specific excitation frequency (Figure 5).

The amplitude of the acceleration signal as a function of time was transformed into a frequency amplitude of the acceleration signal by means of a Fast Fourier Transform (FFT). In this way, it was possible to determine the locations where amplification of the vibration amplitude occurred, indicating the resonance points of the system.

Figure 5
Representation of acceleration amplitude data in time domain (A) and frequency domain-frequency spectrum obtained by Fast Fourier Transform (B).

The average natural frequencies determined in vibration tests were 20.09 Hz for immature stage of maturation and 19.86 Hz for mature maturation stage, with a standard deviation of 0.40 Hz and 0.95 Hz, respectively. For the finite element simulation, the average natural frequencies calculated were 28.41 Hz and 25.56 Hz, for immature and mature stages of maturation, with a standard deviation of 1.04 Hz and 0.33 Hz, respectively (Figure 6).

Figure 6
Results for average experimental and simulated natural frequencies for immature and mature stages of maturation with standard deviation considering the different accessions.

The average values of natural frequencies for each accession at each maturation stage were submitted to the Tukey’s test and it was observed significant differences between experimental and simulated response the 5% of probability (Table 4).

Table 4
Average natural frequencies obtained in vibration tests and models simulation by finite element method for each accession and stage of maturation

	Frequency (Hz)
	Immature				Mature
	BD27	BD40	BGP29	BGP35	BD27	BD40	BGP29	BGP35
Experimental	20.61a	17.81a	19.10a	22.86a	20.99a	17.49a	20.88a	20.10a
Simulated	28.23b	27.83b	30.50b	27.02b	24.41a	27.97b	26.59b	23.17b
Average Error (%)	36.98	56.24	59.67	18.45	16.28	59.92	27.83	15.29
Minimum Error (%)	4.92	12.05	16.70	-3.44	-2.08	20.91	5.28	-1.24
Maximum Error (%)	68.17	80.28	85.72	87.91	67.92	92.92	69.09	48.80

• The averages followed by the same letter, in column, do not differ statistically from each other, by Tukey’s test at 5% probability.

The vibration mode which is associated with the modal shape observed during the experimental vibration tests and is within the measurement range used during the frequency sweep test was characterized as pendulum displacements (Figure 7).

Figure 7
Representation of vibration mode characterized as pendulum displacements.

DISCUSSION

The results of the average natural frequencies suggest that there is a tendency in the frequencies to decay with the evolution in the maturation of the fruits. This behavior is directly related to the increase of mass or rigidity of the system [¹⁵]. According to Velloso and coauthors [⁶], the bunches with fruits at the stage of immature maturation present higher elasticity modulus than at the mature maturation stage, this means that in the mature maturation stage the systems present lower stiffness which causes lower natural frequencies, confirming the direct interference of the maturation in the results obtained here.

There are few studies for biological materials, however in comparative terms, the dynamic behavior observed for fruit-rachilla systems is similar to the coffee fruit-stem system behavior observed by Santos and coauthors, [¹⁰]; Coelho and coauthors [¹⁸] and Tinoco and coauthors [¹⁹]. Santos and coauthors [¹⁰] determined the fundamental frequencies of 23.20 Hz for immature stage of maturation stage and 19.90 Hz for mature stage of maturation stage, using a finite element coffee fruit-stem model for Catuaí Vermelho variety. Additionally, the authors determined for Mundo Novo variety fundamental frequencies of 23.20 Hz and 20.60 Hz for immature and mature stages of maturation, respectively. In addition to the influence of maturation stage, Coelho and coauthors [¹⁸] noticed an influence of the number of fruits present in the system when extracting the natural frequencies, which were among 14.00 Hz and 21.30 Hz for the modes characterized with pendulum displacements. Also for coffee fruit-stem systems Tinoco and coauthors [¹⁹] calculated from the finite element method the fundamental frequencies of 18.68 Hz and 16.33 Hz for the immature and mature stages of maturation, respectively.

Rangel and coauthors [²⁰] worked stochastically to determine the natural frequencies of macaw palm plants. In this approach, the authors used random values for the physical characteristics of the plants, reaching the values of average natural frequencies of 26.05 Hz for the immature maturation stage and 21.23 Hz for the mature maturation stage.

Comparing the cited work with the results obtained, we can see that for the immature stage of maturation the values present greater discrepancy, reaching approximately 22% of difference between them, while for the values in the mature stage of maturation this percentage drops to approximately 6%.

However, the values obtained in the present study corroborate the observed tendency that the natural frequencies decrease with the advance of the maturation of the fruits in the macaw palm bunches.

Villar and coauthors [²⁷] found natural frequency values for macaw palm bunches at the immature stage of maturation, ranging from 26.21 to 33.45 Hz, indicating the high variability of values for the species. This variability is further highlighted by Santos and coauthors. [²⁸].

One way to reach results that are more consistent with reality is to use models that are increasingly representative of the real physical system, as highlighted by Pereira and coauthors [²⁹].

The errors presented in Table 4 represent the level of accuracy of the model. The maximum difference was 92.92% for the access BD40 at the mature maturation stage and the minimum difference was -1.24% for the BGP35 access at the mature maturation stage, this negative sign means that the simulation results underestimated the experimental results.

The average error, calculated from the errors found, was 42.83% for the immature maturation stage and 29.83% for the mature maturation stage, demonstrating that the model showed a best approximation at the mature maturation stage. However in both cases there was an overestimation of the simulated results in comparison to the experimental results.

This behavior can be explained why a mathematical model rarely provides an accurate representation of actual phenomena. In addition the geometric properties may have directly interfered in the deformation characteristics of the studied material. Generally simplified models are used to make numerical simulation feasible, especially in the case of agricultural products in which we work with irregular and variable forms among samples, however these simplifications lead to imprecision of the obtained results. This is a limitation of numerical analysis, however there are some widely used numerical techniques that allow full probabilistic dynamic analysis such as the Duffing equation [³⁴].

The model discretization, the solution algorithm and physical properties of systems employed as input data of the model may also influence the accuracy of the finite element simulations and can be considered an important part of the computer engineering analysis, allowing that simulation results are close to real systems behavior [³⁰].

Pendular and torsional modes are characterized by greater geometric complexity and associated with higher natural frequencies; these mode shapes result in a higher stress generation in the fruit-stem system during the vibration process. According to Tinoco and coauthors [¹⁹], pendular and torsional vibration formscause the connection among rachilla and fruit to constitute a rotational joint that favors the process of detachment of the fruits. From the simulations results, similar modal behavior was observed for macaw palm fruit-rachilla system, also corroborating the results obtained by Rangel and coauthors [²⁰].

Machines that employ mechanical vibrations as an operational principle of work can be adjusted for different vibration parameters in order to detach only fruits in the suitable stage of maturation [¹⁴]. Harvesting process by mechanical vibrations can be applied for detachment of mature fruits, which in some cases increases the quality and, consequently, the market price of the final product, which already happens for the coffee [³¹-³³]. The results of this work will compose a knowledge base for helping machine designers on development of harvest machines for macaw palm fruits by mechanical vibrations that will be necessary to support its productivity chain, improving harvesting efficiency and the quality of the fruits harvested.

CONCLUSION

The methodology used allows one to extract the modal properties of the macaw palm fruit-rachilla system.

Natural frequencies were lower for the mature stage of maturation stage than for immature stage of maturation for vibration tests and finite element simulations. The mean for natural frequencies observed in vibration tests were 20.09 Hz and 19.86 Hz for immature and mature stages of maturation. The mean for natural frequencies, determined by the finite element simulation of the models, were 28.41 Hz and 25.56 Hz, for immature and mature stages of maturation, respectively. The mode shape observed during the vibration tests for the range from 10 Hz to 40 Hz was characterized as pendulum displacements, which was also observed on model simulation.

The results presented here provide data on the dynamic behavior of macaw palm fruit-rachilla system and will serve as a basis for the cropping mechanism design of this crop.

Acknowledgments

The authors thank the Minas Gerais State Agency for Research and Development (FAPEMIG), National Council for Scientific and Technological Development (CNPq) and Coordinating Agency for Advanced Training of Graduate Personnel (CAPES), Brazil, for financial support.

REFERENCES

• ¹
  Ciconini G, Favaro SP, Roscoe R, Miranda CHB, Tapeti CF, Miyahira MAM, et al. Biometry and oil contents of Acrocomia aculeate fruits from the Cerrados and Pantanal biomes in Mato Grosso do Sul, Brazil. Ind. Crops Prod. 2013; 45: 208-14.
• ²
  Azevedo Filho JA, Colombo CA, Berton LHC. Macaúba: Palmeira nativa como opção bioenergética [Macaw Palm: Native palm tree as a bioenergetic option]. Apta Reg. 2012;9:1-10.
• ³
  Roscoe R, Richetti A, Maranho E. [Technical feasibility analysis of oilseeds for biodiesel production in Mato Grosso do Sul]. Rev. Pol. Agr. 2007;16:48-59.
• ⁴
  Carvalho KJ, Souza AL, Machado CC. [Ecology, Management, Forestry and Technology of Macaw Palm]. 2011.[cited 2017 Mar 22]. Available from: http://www.portalmacauba.com.br/2016/10/ecologia-manejo-silvicultura-e.html
  » http://www.portalmacauba.com.br/2016/10/ecologia-manejo-silvicultura-e.html
• ⁵
  Oliveira ZRCR, Santos FL, Valente DSM, Pinto FAC, Velloso NS. Mechanical properties of the rachis from macaw palm bunches. Acta Sci. Agron. 2018;40:1-7.
• ⁶
  Velloso NS, Santos FL, Pinto FAC, Villar FMM, Valente DSM. Mechanical properties of the macaw palm fruit-rachilla system. Pesq. Agropecu. Trop. 2017;47:218-25.
• ⁷
  Villar FMM, Pinto FAC, Santos FL, Queiroz DM, Pereira MR, Valente DSM. Modal properties of the fruit-rachilla system of the macaw palm. PloS One. 2021;16(1):e0237291.
• ⁸
  Brandão AA, Neves JMG, Silva HP, Continho PH, Aquino CF, Santos PA, et al. [Biometric characterization of macaw palm fruits at different stages of maturation, from two regions of the state of Minas Gerais]. Glob. Sci. Technol. 2014;7:15-23.
• ⁹
  Sanjinez-Argandoña EJ, Chuba CAM. [Biometric, physical and chemical characterization of fruits of the bocaiuva palm Acrocomia aculeata (Jacq) Lodd]. Rev. Bras. Frutic. 2011;33:1023-8.
• ¹⁰
  Santos FL, Queiroz DM, Valente DSM, Coelho ALF. Simulation of the dynamic behavior of the coffee fruit-stem system using finite element method. Acta Sci. Technol. 2015;3:11-7.
• ¹¹
  Sola-Guirado RR, Castro-García S, Blanco-Roldán GL, Jiménez-Jiménez F, Castillo-Ruiz F J, Gil-Ribes JA. Traditional olive tree response to oil olive harvesting technologies. Biosyst. Eng. 2014;118:186-93.
• ¹²
  Castro-García S, Blanco-Roldán GL, Gil-Ribes JA. Frequency response of Pinus Pinea L. for selective cone harvesting by vibration. Trees. 2011;25:801-8.
• ¹³
  Reddy JN. An introduction to the finite element method. Singapore: McGraw-Hill International Editions; 1993. 684p.
• ¹⁴
  Grupioni CMF, Santos FL, Fernandes HF, Valente DSM, Pinto FAC. Development and evaluation of operational performance of macaw fruits semi-mechanized harvester by means of mechanical vibrations principle. Semina: Cienc. Agrar. 2018;39(2):497-510.
• ¹⁵
  Grupioni CMF, Santos FL, Velloso NS, Valente DSM, Pinto FAC. Macaw palm supply chain: Evaluation of a semi-mechanized fruit harvesting system. Ind. Crops Prod. 2020;151:e112444.
• ¹⁶
  Rao S. [Mechanical Vibrations]. São Paulo: Pearson, 2008.448 p.
• ¹⁷
  Zienkiewicz OC, Taylor RL, Zhu JZ. The finite element method: its basis and fundamentals. Elsevier Butterworth-Heinemann, 2005. 733p.
• ¹⁸
  Coelho ALF, Santos FL, Queiroz DM, Pinto FAC. Dynamic behavior of the coffee fruit-stem-branch system using stochastic finite element method. Coffee Sci. 2016;11:1-10.
• ¹⁹
  Tinoco HA, Ocampo DA, Peña FM, Sanz-Uribe JR. Finite element modal analysis of the fruit-peduncle of Coffea arabica L. var. Colombia estimating its geometrical and mechanical properties. Comput. Electron. Agric. 2014;108,17-27.
• ²⁰
  Rangel JP, Queiroz DM, Pinto FAC, Teixeira CC, Santos FL, Valente DSM. Dynamic behavior of the macauba palm (Acrocomia aculeata) fruit-rachilla system using the stochastic finite element method. Acta Sci. Agron. 2021;43:e48565.
• ²¹
  Ewins DJ. Modal Testing: theory, practice and application. Research Studies Press Ltd., 2000.576p.
• ²²
  Baharin NH, Rahman RA. Effect of accelerometer mass on thin plate vibration. J. Mekanikal. 2009;29:100-11.
• ²³
  Boyce WE; Diprima RC. [Elementary Differential Equations and Boundary Value Problems]. Rio de Janeiro: Editora LTC, 2002.416p.
• ²⁴
  Hoffman JD. Numerical Methods for Engineers and Scientists. McGraw-Hill Inc., 1992. 801p.
• ²⁵
  National Instruments. Lab VIEW Graphical Programming for Instrumentation, Austin, Versão 5.0, 1998.
• ²⁶
  R Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. 2017. Available from: https://www.R-project.org/.
  » https://www.R-project.org
• ²⁷
  Villar FMM, Pinto FAC, Santos FL, Grossi JAS, Velloso NS. Elasticity modulus and damping ratio of macaw palm rachillas. Cienc. Rural. 2017;47:1-6.
• ²⁸
  Santos FL, Scinocca F, Marques DS, Velloso NS, Villar FMM. Modal properties of macaw palm fruit-rachilla system: An approach by the stochastic finite element method (SFEM). Comput. Electron. Agric. 2021;184:e106099.
• ²⁹
  Pereira MR, Santos FL, Velloso NS, Villar FMM, Rodrigues MR. Modeling and simulation of the dynamic behavior of the macaw palm fruit-rachilla system. Simulation. 2021;1:1-14.
• ³⁰
  Celik HK. Determination of bruise susceptibility of pears (Ankara variety) to impact load by means of FEM-based explicit dynamics simulation. Postharvest Biol. Technol. 2017;128:83-97.
• ³¹
  Tinoco HA, Peña FM. Harmonic stress analysis on Coffea arabica L. var. Colombia fruits in order to stimulate the selective detachment: A finite element analysis. Simulation. 2017;1:1-12.
• ³²
  Silva FC, Silva FM, Alves MC, Ferraz GAS, Sales RS. Efficiency of coffee mechanical harvesting in different vibration during harvest time. Coffee Sci.2015;10:56-64.
• ³³
  Castro-García S, Blanco-Roldán GL, Gil-Ribes JA, Agüera-Vega J. Dynamic analysis of olive trees in intensive orchards under forced vibration. Trees. 2008;22:795-802.
• ³⁴
  Kamiński M, Corigliano A. Numerical solution of the Duffing equation with random coefficients. Meccanica. 2015;50:1841-53.

• Funding:

  This research received no external funding.

Edited by

Editor-in-Chief:

Alexandre Rasi Aoki

Associate Editor:

Aline Alberti

Publication Dates

• Publication in this collection
  08 Nov 2024
• Date of issue
  2024

History

• Received
  10 June 2022
• Accepted
  14 Aug 2024