Three-Dimensional Scanning and Computational Simulation of Coffee Tree Branches

Abstract

Coffee has significant economic importance in Brazil, which is the world’s largest exporter of this product. The main focus of this work is to obtain a computational model of coffee branches, using scanning techniques and experimental validation through modal analysis, to understand their mechanical behaviors due to dynamic and static properties. The main mechanical properties of the branches, including specific gravity and elastic modulus, were obtained in the laboratory and used as input parameters for the simulations. Geometric models were obtained by means of three-dimensional scanning and analyzed using the finite element method to predict the static and dynamic behaviors of the branches. The methodology was validated in experimental tests. The results of vibration frequencies obtained in the branches showed variations between simulations and experimental tests ranging from 2% to 20%, and the statistical analysis did not find significant differences. The results demonstrate important indicators for predicting the static and dynamic behaviors of plagiotropic branches of a coffee tree. These results can be considered in vibration analysis for coffee harvesting, which represents a technological advance in the area, since simplified models are usually obtained by geometric coordinates to obtain geometric models of a coffee tree.

1. Introduction

Brazilian coffee is appreciated worldwide because the diversity of climates, reliefs, altitudes, and latitudes where the coffee trees are grown favors the production of different types of beans, meeting national and foreign demands and favoring blends [1]. The country is the largest coffee producer and exporter, and according to the National Supply Company, CONAB, Brazil produced 50.9 million and exported 39.8 million 60-kg bags in 2022 [2]. Given this high production, harvesting procedures are in need of improvement to replace, to the extent possible, human labor with mechanized labor.

Among the machinery used in harvesting processes, those that efficiently vibrate plant stems for agricultural products such as olives and nuts are also used in coffee harvesting [3,4]. However, the poor timing of coffee plant vibration can lead to branch breaks and/or plant stress with damage, such as defoliation, which compromises the production of subsequent beans because the plant will use its energy reserves to replace the lost leaves instead of the fruits [3]. To maintain coffee plant integrity, the vibration parameters must be applied at a frequency and time sufficient for the fruits to loosen without causing damage to the branches [5].

With advances in computer modeling and simulation, numerical methods have been used to study dynamic systems, such as the Finite Element Method (FEM), which has already been applied in agricultural engineering studies. Li, Andrews, and Wang [6] developed a model capable of estimating the response of tomatoes to applied forces and thus determining the influence of postharvest management on fruit damage. The FEM is a practical method for solving complex systems because it simplifies the problem and reduces the number of equations by measuring the local effects of point nodes. When subdividing (discretizing) the domain into smaller geometric elements, the so-called finite element mesh, its solution is represented by continuous functions at the nodes of each element, which are measured to represent the entire problem of the regional domain [7]. Tian et al. [8] also applied FEM to simulate the mechanical behavior and internal damage of kiwifruit under compressive loads, demonstrating the feasibility of using this method to analyze internal stress and deformation. When modeling plants such as coffee trees, the FEM can be applied to evaluate their dynamic behavior when vibration is applied during mechanized harvesting.

The application of numerical methods, such as the Finite Element Method (FEM), has proven to be a powerful tool for understanding the mechanical behavior of machines and coffee plants during harvesting processes [9,10,11,12,13,14,15]. Pereira et al. [16] provided three-dimensional models of coffee fruits at different ripeness stages for fruit compression simulations. The numerical analysis revealed that ripe fruits exhibited higher deformation and lower von Mises stress compared to unripe fruits, highlighting how maturation stage directly influences susceptibility to mechanical damage. This type of simulation allows for adjustments in harvesting machinery to reduce losses caused by fruit bruising and breakage.

In the context of the dynamic behavior of coffee plant structures, Melo et al. [17] applied FEM to analyze plagiotropic branches of the Catuaí Vermelho variety, modeling their natural frequencies and vibration modes based on experimentally obtained physical and mechanical data. The models were validated through modal analysis, showing agreement between simulated and laboratory-measured results. This approach is essential for the development of vibration-based harvesting systems, as it enables the identification of excitation frequencies that optimize fruit detachment while minimizing damage to the plant structure.

Additionally, numerical simulation techniques have been successfully used to understand and predict mechanical stresses in other crops, as shown in the study by Santos et al. [18] on sugarcane basal cutting. Using FEM, the authors modeled the cutting process by considering the anisotropic nature of the plant, emphasizing the importance of distinguishing between skin and pulp regions to accurately simulate stress distributions. This reinforces the need for precise modeling in coffee cultivation as well, especially when simulating the complex interactions between the fruit, branches, and harvesting mechanisms.

In this sense, it is essential to study the interaction between the plant and the harvester, as well as the static and dynamic behavior of the plagiotropic and orthotropic branches, to understand the frequencies, charges, and amplitudes necessary to detach the coffee fruit without damaging the coffee tree. Orthotropic branches are structures that grow vertically from the main stem, contributing to the overall growth of the plant. In contrast, plagiotropic branches grow horizontally and are primarily responsible for fruit production, being more susceptible to the impact of vibrations during mechanical harvesting.

This type of modal analysis of the properties of the fruit-twig system is crucial in the development of agricultural mechanization [19]. Ferreira Júnior et al. [20] studied the vibrational displacement of coffee branches, and by processing the signals obtained, it was possible to estimate the dynamic displacements performed by the branches at harvest.

Regarding simulations, the closer a model is to the real system, the more accurate its representation will be, and the use of reverse engineering is a great approach because it is an inverse process starting from the physical model and ending in computer-aided design, or CAD [21]. The practice of scanning is an interesting solution since the generation of 3D virtual models expresses with great accuracy the geometry of an object, especially of organic elements, which have variations, even within the same species [22]. The use of images obtained by scanning has been described for soil texture analysis [23] and fruit damage studies [20] and applied to coffee studies [19].

In this study, the physical and mechanical properties of coffee branches were experimentally extracted and digitized, followed by their computational reconstruction and subsequent submission to dynamic analyses using the Finite Element Method (FEM). This procedure was conducted with the purpose of comparing the calculated dynamic properties with those obtained experimentally. The primary objective was to assess whether the real and complex shape of the organic structure can yield simulation results sufficiently close to the experimental data. We demonstrate that this approach paves the way for the development of more robust computational models, which may in the future support the design and optimization of agricultural implements.

2. Theoretical Framework for Dynamic Analysis of Coffee Branches

The dynamic behavior of structures, such as coffee tree branches subjected to external forces (e.g., from mechanical harvesters), can be effectively analyzed using the Finite Element Method (FEM). This approach allows the complex geometry of the branch to be discretized into a finite number of smaller, simpler elements connected at nodes. The continuous system’s behavior is then approximated by the collective behavior of these elements, described by a system of differential equations [24].

The general equation of motion for a linear, multi-degree-of-freedom (MDOF) system, representing the discretized coffee branch via FEM, can be expressed in matrix form (Equation (1)).

$$\left[m\right]\left\{\mathrm{ẍ}\right(t\left)\right\}+\left[c\right]\left\{\mathrm{ẋ}\right(t\left)\right\}+\left[k\right]\left\{x\right(t\left)\right\}=\left\{F\right(t\left)\right\}$$

(1)

where:

• $\left[m\right]$ is the system mass matrix, derived from the mass properties of the finite elements.
• $\left[c\right]$ is the system damping matrix, representing energy dissipation mechanisms within the branch material and at connections.
• $\left[k\right]$ is the system stiffness matrix, derived from the elastic properties and geometry of the finite elements.
• {$x\left(t\right)\},$ $\left\{\mathrm{ẋ}\right(t\left)\right\},$ and $\left\{\mathrm{ẍ}\right(t\left)\right\}$ are the vectors of nodal displacements, velocities, and accelerations, respectively, as functions of time t.
• $\left\{F\right(t\left)\right\}$ is the vector of externally applied nodal forces, such as those simulating harvester impacts or shaker excitation.

To understand the inherent dynamic characteristics of the coffee branch, modal analysis is performed. This involves determining the natural frequencies at which the structure tends to vibrate freely and the corresponding mode shapes (patterns of deformation). These properties are fundamental as they dictate how the branch will respond to dynamic loading, particularly near resonance.

Modal analysis typically focuses on the undamped, free vibration case, where external forces and damping are neglected [24,25,26]. Therefore, Equation (1) is simplified to Equation (2).

$$\left[m\right]\left\{\mathrm{ẍ}\right(t\left)\right\}+\left[k\right]\left\{x\right(t\left)\right\}=\left\{0\right\}$$

(2)

Assuming a harmonic solution of the form {x(t)} = {Φ}e^{iωt}, where {Φ} is the vector of vibration amplitudes (mode shape), ω is the angular natural frequency (rad/s), and i is the imaginary unit (√−1), substituting this into Equation (2) leads to the generalized eigenvalue problem, resulting in Equation (3).

$$\left(\right[k]-\omega 2[m\left]\right)\left\{\Phi \right\}=\left\{0\right\}$$

(3)

Solving this eigenvalue problem yields a set of eigenvalues, ${\omega }_{\mathrm{i}}2$, and corresponding eigenvectors, ${\left\{\Phi \right\}}_{\mathrm{i}}$, where the square roots of the eigenvalues, ${\omega }_{\mathrm{i}}$, are the angular natural frequencies of the system. These are often converted to ordinary frequencies $f_{\mathrm{i}}={\omega }_{\mathrm{i}}/\left(2\pi \right)$ (Hz). The eigenvectors, $\left\{\Phi \right\}$_i, represent the mode shapes, describing the pattern of relative displacement of the nodes for each natural frequency. These natural frequencies and mode shapes are crucial outputs from the FEM simulation, providing insight into the potential resonant behavior of the coffee branch.

While modal analysis identifies the inherent frequencies and shapes, Frequency Response Function (FRF) analysis characterizes the branch’s response to a frequency-dependent input force. Experimentally, this is often achieved by applying an impact force (broad frequency content) and measuring the resulting acceleration at specific points. Computationally, it involves solving Equation (1) in the frequency domain.

The FRF, often denoted as $H\left(\omega \right),$ represents the ratio of the system’s output response (e.g., displacement or acceleration at a point) to the input force (at the same or another point) as a function of angular frequency $\omega$. Peaks in the magnitude of the FRF typically correspond to the system’s natural frequencies (${\omega }_{\mathrm{i}}$) identified through modal analysis. The height and width of these peaks are influenced by the system’s damping $\left(\right[c]$), which, while neglected for calculating ${\omega }_{\mathrm{i}}$ and ${\left\{\Phi \right\}}_{\mathrm{i}}$ in the eigenvalue problem (Equation (3)), is present in the real structure and affects its actual response amplitude and energy dissipation.

In this study, FEM simulations are used to predict the dynamic characteristics (${\omega }_{\mathrm{i}}$, ${\left\{\Phi \right\}}_{\mathrm{i}}$) via Equation (3), and these predictions are validated against experimentally measured FRFs obtained from impact testing on actual coffee branches. Comparing the simulated natural frequencies with the peaks observed in the experimental FRFs allows for model validation and refinement.

Coffee branches, being biological materials, possess complex geometries and material properties that can introduce challenges. While this study primarily employs linear FEM analysis (Equation (1) assumes linear elasticity and small displacements), it is acknowledged that wood can exhibit anisotropic and potentially non-linear behavior under certain loading conditions. The intricate branching structure also contributes to the complexity. The scanning technique aims to capture the geometric details accurately, while the linear FEM provides a fundamental understanding of the dynamic response, serving as a basis for comparison with experimental data and potential future non-linear investigations.

3. Materials and Methods

3.1. Collection and Selection of the Branches

The coffee plants used were twelve-year-old and repruned four years ago, belonging to the family Rubiacea, genus Coffea arábica L. (Carl Linnaeus) Catuaí-Vermelho (IAC 144), which were provided by the Center for Studies in Coffee Culture of the Federal University of Lavras and collected in June of 2023. This area has a humid subtropical climate, with rainy summers and dry winters, and an average annual temperature ranging from 20 to 22 °C. The coffee plants were fertilized based on soil analysis and received regular cultural practices such as pruning, phytosanitary control, and occasional supplemental irrigation. The selected samples were in good phytosanitary condition, showing no visible signs of pests, diseases, or water stress.

Three branches were randomly collected from the middle third of the coffee trees by sawing between the orthotropic and plagiotropic branches. Although leaves and fruits were not used in the simulations, they were carefully removed during scanning to preserve their insertion bases. The peduncle and the beginning of the leaf structure were preserved for reliable scanning, and the anatomical markers left by these insertions allow digital reassembly of the branch, its leaves, and fruits in their original positions and angles in future studies.

The flow of activities and experiments performed to obtain the results is shown in Figure 1 and the plagiotropic branch cutting step sequence is shown in Figure 2.

3.2. Scanning of Plagiotropic Branches

The fruits and leaves were removed, and digitalization was performed as shown in Figure 3. The branches were clamped in a bench vise using a 7-axis FARO Quantum FaroArm^® Series 3D portable measuring arm (FARO Technologies, Lake Bluff, IL, USA) associated with a QUANTUMS SCANARM HD scanner plug and play type with an accuracy of 25 μm and a maximum acquisition capacity of 1.2 million points per second.

For digitization, the software Polyworks MS^® v.2022 (InnovMetric Software Inc., Québec, QC, Canada) was used. The generated file comprised all the scanned meshes, which were treated later. The scans were saved with different numbers for the respective branches.

After the branches were digitized, all generated meshes were reworked using Polyworks ${Modeler}^{\circledR }$ v.2022 (InnovMetric Software Inc., Québec, QC, Canada) to remove imperfections and gaps arising from the digitization process. The result was a polygonal surface model by means of STL (Standard Triangle Language) file, as shown in Figure 4.

3.3. Validation of the Scanning Methodology

To validate the method for obtaining the model from the scanner, a specimen obtained from the orthotropic branch was used. Although the test was nonstandard, the specimen was cut and worked until it obtained the appearance shown in Figure 5. This single-sample preliminary test aimed exclusively to assess the correlation between experimental results and finite element simulations using the real scanned geometry, thereby ensuring the reliability of our overall approach.

The specimen was scanned using PolyWorks ${Inspector}^{\circledR }$ v.2022 (InnovMetric Software Inc., Québec, QC, Canada), generating a point cloud that, after treatment using PolyWorks ${Modeler}^{\circledR }$ v.2022 (InnovMetric Software Inc., Québec, QC, Canada), obtained a polygonal STL model, which, after reworking, was generated in an IGES (Initial Graphics Exchange Specification) file, which was used for the simulations.

The specimen was subjected to a tensile test using an Instron^® EMIC 23-20 universal testing machine (INSTRON Inc., Norwood, MA, USA) equipped with a maximum load cell of 20 kN, with a preload of 20 N and a 1 mm feed/min, until failure. The results of this test were used for the experimental identification of the elastic modulus and as input data for the static simulation.

The specific gravity of the specimen was obtained using the relationship presented in Equation (4). The mass was obtained experimentally using a Gehaka AG200 balancer (GEHAKA Inc., São Paulo, SP, Brazil)with a precision of 0.0001 g, and the volume was obtained using the polygonal model in Polyworks Modeler^® software v.2022 (InnovMetric Software Inc., Québec, QC, Canada),

$$\rho ={\displaystyle \frac{m}{V}}$$

(4)

The physical–mechanical properties of the material obtained experimentally were used as input parameters for Ansys^® v. 14.5 (Ansys Inc., Canonsburg, PA, USA) in the static simulations of stress and deformation.

For the simulations in the specimen model, computational tensile tests were performed using the same parameters as the experimental tensile test up to a load of 4620 N in a total time of 216 s. The Poisson’s ratio used was determined by Velloso et al. [5] because it was a specimen from an orthotropic branch of coffee wood, with a value of 0.25. The specimen was considered an isotropic material, limited to deformation in the transverse and rotational axes. The base was fixed, and the force was applied to the top of the specimen, as shown in Figure 6.

Prior to the simulations, mesh convergence was performed to ensure the numerical stability of the results and mitigate the oscillations resulting from the discretization characteristics of the finite element model.

3.4. Experimental Modal Analysis

After scanning the branches, vibration analyses were conducted using a Piezotronics™ PCB 086C03 impact hammer equipped with a force sensor and a Piezotronics™ PCB 352C33 accelerometer (PCB Pezotronics Inc., Depew, NY, USA), whose mass accounted for less than 10% of the total system mass, as shown in Figure 7. The accelerometer was mounted vertically at one-third of the total branch length, while impacts were applied at the midpoint of each branch. Ten impacts were performed on each branch to ensure data reliability and repeatability. Acceleration signals from both the accelerometer and the impact hammer were collected using a National Instruments™ NI cDAQ-9174 data acquisition system (National Instruments™ Inc., Austin, TX, USA). The acquired data were processed using LabView^® software with the Sound and Vibration package, and frequency response function parameters were extracted using the MATLAB v. 9.13.0 R2022b Student version, (Natick, MA, USA).

Next, the branches were partitioned and worked until a square profile with 4 mm edges and a length of at least 5 times its edge was reached. The tensile tests were performed to define the elastic modulus (Young’s modulus) using Hooke’s law, according to Equation (5).

$$\sigma =E\times \epsilon$$

(5)

where $\sigma$ is the stress applied to the material (Pa); $E$ is the material’s modulus of elasticity, also known as Young’s modulus (Pa), and $\epsilon$ is the relative deformation or elongation of the material (a dimensionless measure).

The test was performed using an Instron^® EMIC 23-20 universal testing machine equipped with a maximum load cell of 20 kN, as shown in Figure 8a. This equipment works together with BLUEHILL Testing software v. 3.68.0.4518 (Instron Inc., Norwood, MA, USA). In the machine, the branches were positioned parallel to the fibers, fixed by claws at the ends, and subjected to loading in the central region, as shown in Figure 8b.

A preload of 20 N was applied for tensioning and positioning, ensuring that the jaw clearances and the specimen were well adjusted at a displacement rate of 10 mm/min with a time duration corresponding to the linearity exit after rupture of the specimen. The specific gravity of the wood was obtained by the Archimedes immersion method, also with the pieces of the branches.

3.5. Modeling, Mesh Convergence, and Simulations

Ansys^® version 14.5 was used to process the dynamic analysis of the model. The experimental mechanical properties of the material were entered, and the simulation was performed by extracting the vibration modes and their respective frequencies. The convergence of the mesh was obtained by its refinement based on the percentage difference in relation to the result of the previous mesh Ansys^® version 14.5 was used to process the dynamic analysis of the model. The experimental mechanical properties of the material were entered, and the simulation was performed by extracting the vibration modes and their respective frequencies. The convergence of the mesh was obtained by its refinement based on the percentage difference in relation to the result of the previous mesh. For the finite element modeling, the SOLID187 (TET10) element was used, a ten-node quadratic tetrahedral element suitable for modeling complex organic geometries, such as scanned branches. Mesh refinement was conducted with element sizes of 5 mm, 1 mm, 0.5 mm, and 0.25 mm. Mesh convergence was considered acceptable when the variation in the third natural frequency between successive refinements was less than 0.1%. Figure 9 illustrates the finite element mesh generated for one of the scanned branches using SOLID187 (TET10) elements, highlighting the high mesh density required to accurately represent the complex organic geometry.

3.6. Validation of the Scanning Method

The specimen of the material, collected from an orthotropic branch of a coffee tree, was subjected to an experimental tensile test. The experimental data, including the applied load and the resulting deformation, were carefully recorded and treated to establish the physical properties of the material. A three-dimensional numerical model of the same specimen was created using 3D scanning, mesh correction techniques, and the mechanical properties of the material. The loading conditions and constraints were replicated according to the experimental test.

As simulation parameters, the force and time data obtained by the universal testing machine, as shown in Figure 10, were used, and the stress and displacement results were compared.

Considering the isotropic material, the loading and constraints used in the experimental test were then reproduced in the numerical model, aiming at the virtual recreation of real conditions. The Poisson’s ratio used in the simulation was that found in the literature, provided by Velloso et al. [5] and set at 0.25 for orthotropic branches and 0.09 for plagiotropic branches. The specimen in question had a specific mass of 780 kg/m³. The tensile test performed revealed a maximum stress of 37.642 MPa and a maximum displacement of 4.755 mm.

4. Results and Discussion

4.1. Validation of the Three-Dimensional Scanning Process

The elastic modulus was determined from the line of slope secant to the curve (dashed in red), which relates the stress as a function of the specific deformation (Figure 11a). This straight line was defined based on the points located at 10 and 50% of the tensile strength, parallel to the fibers, following the guidelines prescribed by ABNT NBR 7190 [27]. The tensile test allowed the inference of the elastic modulus of the material, and its estimate was calculated as 447.89 MPa.

In the analysis of the mechanical properties, both the numerical simulations and the experimental tests identified a linear–elastic behavior, where the stress versus strain graph can be considered linear, with an R² of 0.99713, as shown in Figure 11b.

Regarding the computer simulation, considering the experimental data as input, the mean stress was obtained at 43.006 MPa, with a maximum stress of 144.94 MPa and a maximum displacement of 4.7425 mm.

The mesh used in the computer simulations of the validation of the specimen is shown in Figure 12.

Considering the equivalent stress, with a mesh element size of 0.5 mm, the results converged from 113,347 elements, presenting a coefficient of determination of 98.2%, as shown in Figure 13 and

Table 1. Specific and statistical mass of the branches.

Equivalent Stress (MPa)	Relative Error (%)	Nodes	Elements
101.48		9764	5750
118.52	15.943	17,760	11,073
138.12	15.277	56,721	38,528
142.34	3.008	95,464	65,890
144.94	1.886	161,291	113,347

.

Thus, it can be concluded that the mesh converges to a satisfactory result, showing an average stress with a variation of 12.47% of the experimental result and only 0.02% in the deformation.

The simulation was performed on a Dell G15 5111 notebook with an 11th-generation Intel(R) Core (TM) i5-11400H processor with a frequency of 2.70 GHz, 24 GB of RAM Memory (Random Access Memory), a frequency of 3200 MHz, and an Nvidia RTX 3050 video card. The static simulation used 0.77 GB of RAM, using 2.8 GB of disk, and completed the simulation in 3 min 51 s.

A simplified model was tested to assess the computational cost of the scanned model, which differs only because it is a flattened model, using modeling to remove imperfections such as wood knots, inserts, and other material defects. In this case, the use of RAM was 0.34 GB, and the disk space used was 0.7 GB, with a finalization time of 1 min 10 s.

Thus, it is clear that the addition of the material characteristics directly affects the computational cost, where the memory use was 57.34% higher and the disk space used was 72.50% higher than the simplified model. The simplified model was not able to demonstrate a satisfactory result, as the model was based on scanning, using the data of the experimental results as parameters.

The voltage variation can be attributed to the scanning process, which reveals the imperfections present in the material. These imperfections result in stress concentration points. This effect can be observed both in the experimental model (Figure 14a) and in the simulated model (Figure 14b). The collapse site of the experimental specimen was the same and presented the maximum stress.

4.2. Physical and Mechanical Properties of the Branches Used in the Simulations

The specific mass of the branches and the statistical study are presented in

Table 2. Specific and statistical mass of the branches.

Branches	Average (kg/m3)	Minimum Value (kg/m3)	Maximum Value (kg/m3)	Coefficient of Variation (%)
1	795.31	750	840	5.14
2	896.33	695	1340	31.40
3	722.67	533.33	940	24.79

.

The average specific gravity obtained by Velloso et al. [5] for the plagiotropic branches showed a mean value of 1036.33 kg/m³. To understand the range in which the genuine population mean could reside, a confidence interval with a confidence level of 95% was applied. Consequently, the calculations delineated an estimated range between 840.88 and 1237.78 kg/m³, a range that basically comprises all the data found, while Coelho et al. [28] also conducted a study of properties, and their observed metrics presented a mean specific mass value of 900 kg/m³ and a standard deviation of 110 kg/m³ and delineated an estimated range between 681.4 and 1118.6 kg/m³ for the true mean, which also comprises most of the data found in the present study.

The elastic modulus was 655.52 MPa for R1, 600.63 MPa for R2, and 580.40 MPa for R3, and compared to other studies, such as Velloso et al. [5], the average modulus of elasticity obtained for these plagiotropic branches was 507.72 MPa, while Carvalho et al. [19] presented a value of 2041.5 ± 326.1 MPa. However, in this study, there was no differentiation between orthotropic and plagiotropic branches, and the sample was made from the orthotropic branch of the plant with a compression test, which may explain the large variation in the results of the present study.

The choice to adopt a preexisting value from the literature is due to the complexity of directly obtaining the Poisson’s ratio of a specific material or component, in addition to demanding significant financial resources, which made this procedure unfeasible. In addition, Poisson’s ratio values may vary depending on the test conditions and material characteristics, making the estimate based on the literature a practical alternative.

To validate the mesh convergence of the model, the meshes in question were subjected to analyses at the third natural frequency. This procedure was adopted to determine the convergence of the results in relation to the data obtained through numerical simulations. An iterative process was implemented, consisting of the successive comparison of the results obtained in each iteration, to achieve satisfactory convergence. This attests to the reliability of the results, ensuring that they are in line with the parameters of interest.

4.3. Modeling, Convergence Analysis, and Branch Simulation

The modeling of the three branches was performed with the aid of Faro^® arm equipment, and the scanning processing was performed by Polyworks^® Inspector and Modeler^® software, which generated STL files with high representation compared to the real model. The data of the models are presented by branch, according to Figure 15, Figure 16 and Figure 17 and

Table 3. Statistical data of the R1 model.

Number of points	363,288
Average deviation	0.022 mm
Standard deviation	0.064 mm
Surface out of tolerance	0.265%

,

Table 4. Statistical data of the R2 model.

Number of points	386,746
Average deviation	0.006 mm
Standard deviation	0.061 mm
Surface out of tolerance	0.004%

and

Table 5. Statistical data of the R3 model.

Number of points	212,432
Average deviation	0.086 mm
Standard deviation	0.147 mm
Surface out of tolerance	0.708%

, and present the number of points collected in each model in the scan and the statistical data of deviations with a maximum tolerance of 0.5 mm after processing for conversion to the IGES model.

4.4. Mesh Convergence Analysis of the Branches

In sample R1, when using elements with dimensions of 5 mm, a natural frequency of vibration of 25.953 Hz was observed using a mesh composed of 2472 elements. However, when performing a refinement with 1 mm elements, the frequency decreased to 24.414 Hz, resulting in a decrease of 2.16%. Importantly, this refinement was performed based on a significant increase in the mesh size, which grew by more than 420%. For elements with dimensions equal to 0.5 mm, a frequency of 25.396 Hz was obtained, with a decrease of 0.073% compared to the previous value. However, this improvement in resolution was accompanied by a substantial increase in the mesh density, which increased by 424.90%. Further refinement for 0.25 mm elements resulted in a frequency of 24.38 Hz, with a decrease of 0.020%; however, the mesh density increased considerably, reaching an increase of 420.34% for branches R2 and R3. The aspects of convergence for branches R1, R2, and R3 are presented in

Table 6. Mesh convergence data and computational cost of R1, R2, and R3.

	Sise of Mesh Elements (mm)	Number of Mesh Elements	Memory Used (Gb)	Disc Used (Gb)	Time	3rd Vibration Mode (Hz)	Coefficient of Variation (%)
R1	5	2472	0.16	0	00:04	24.953
	1	129,449	7.25	0.9	00:26	24.414	2.16
	0.5	679,582	4.27	5.8	05:00	24.396	0.073
	0.25	3,536,191	17.73	37.7	35:40	24.390	0.024
R2	5	13,838	0.16	0	00:04	9.988
	1	194,101	2.16	6.5	01:40	10.112	1.23%
	0.5	1,780,485	12.98	19.5	40:28	10.1	0.11%
	0.25	5,224,086	26.36	47.5	29:55	10.111	0.10%
R3	5	7989	0.43	0.1	00:03	15.405
	1	148,244	8.26	1.0	00:26	15.353	0.33%
	0.5	800,462	4.67	6.4	04:49	15.363	0.060%
	0.25	3,345,365	19.94	28.1	26:58	15.444	0.52%

. Computational costs are presented in Figure 18, Figure 19 and Figure 20.

The mesh convergence results of the branches attest to the need for a balance between resolution and computational efficiency when performing mesh refinements in FEM simulations. However, it is possible to note that all results converge, without significant discrepancies, especially due to the considerable variability of the geometry, and that the responses converge satisfactorily, with low coefficients of variation.

4.5. Experimental Modal Analysis of the Branches

From the analysis of the frequency response functions (FRFs), the natural frequencies of the system were determined for the first branch (R1). The data obtained from the FRF went through preprocessing by converting the magnitude to the decibel scale (dB) and the application of a high-pass filter to highlight the frequencies with higher amplitudes while attenuating the frequencies whose amplitude was below the 5 dB threshold.

The processed data were then graphically represented, as shown in Figure 21, disregarding data with coherence lower than 80%, and the representative peaks were identified using MATLAB^® software. However, an irregularity in the FRF curve was observed after the third peak, with the absence of prominent peaks. In this context, we chose to select the first three natural frequencies, corresponding to the three initial peaks identified.

Regarding the second branch (R2), the geometry diverged from the geometry of R1, and consequently there were differences in their natural frequencies. Notably, R2 presented two second-order plagiotropic branches.

An irregularity in the curve was also observed from the third peak, as shown in the graph of Figure 22.

Branch 3 (R3) also had two second-order plagiotropic branches, one of which had fractured before its removal from the coffee tree, and the FRF response obtained is shown in Figure 23.

The FRF acquisition process involved the application of impacts in the central part of the branch of interest, following an acceptance/rejection criterion based on the coherence obtained from 10 effective impacts. Coherence is a metric that quantifies the relationship between the power of the response signal and the power of the input signal. In essence, it can be compared to the correlation coefficient but applied in the frequency domain and serves as a measure of the presence of noise in the acquired signals.

Notably, due to the use of only one accelerometer during the acquisition process, the data obtained are related to the frequencies on the vertical axis. This approach allows the evaluation of the system response to excitations at specific frequencies, contributing to the characterization of the dynamic properties of the branches in question.

Melo et al. [29] determined the frequencies using a similar data acquisition methodology, for which they obtained, in the first three modes for the middle branches, the natural frequencies of 6.40 Hz, 21.63 Hz, and 44.93 Hz. The relative differences between the frequencies obtained for the first three vibration modes are presented in

Table 7. Percentage of difference in natural frequencies (Hz) between the data obtained by Melo et al. [29] and the present study.

	Melo et al. [29]	R1	Relative Difference R1	R2	Relative Difference R2	R3	Relative Difference R3
1st mode	6.40	3.072	52.03%	3.072	52.03%	4.096	36.09%
2nd mode	21.63	10.240	52.65%	5.120	76.32%	8.192	62.13%
3rd mode	44.93	31.744	29.34%	14.336	68.08%	14.336	68.08%

, and the difference between the natural frequencies can be explained by obtaining the geometry of the branches. While Melo et al. [29] used models obtained from coordinate images, this study obtained the images by scanning.

4.6. Computer Simulations of the Branches

In the case of branch R1 (Figure 24), the first natural frequency showed a negative percentage difference of −20.022%, the second frequency of −2.19%, and the third frequency of −23.17% in relation to the values obtained experimentally.

The analyses of branch R2 (Figure 25). The first natural frequency showed a positive percentage difference of 7.018%, the second frequency of 7.34%, and the third frequency of −9.88%, compared to the experimental results.

In relation to the R3 branch (Figure 26), the first natural frequency showed a percentage difference of −15.58%, the second frequency of −9.19%, and the third frequency of 2.62%, in relation to the experimental values.

The results of the experimental frequencies (exp.) compared to the computational results (comp.) of the branches are presented in

Table 8. Comparison of the natural frequencies in Hz of the experimental and simulated results for R1, R2, and R3, and relative error (RL) in percentual (%).

Natural Freq.	R1 Simul.	R1 Exp.	R1 RL	R2 Simul.	R2 Exp.	R2 RL	R3 Simul.	R3 Exp.	R3 RL
w1	2.456	3.072	25.08	3.287	3.072	6.99	3.457	4.096	15.6
w2	10.016	10.24	2.18	5.495	5.12	7.32	8.346	9.192	9.20
w3	24.39	31.744	23.16	14.336	15.908	9.88	14.713	14.336	2.62

.

In studies conducted by Melo et al. [29], the natural frequencies for branches without leaves and without fruits were defined according to

Table 9. Natural frequencies obtained by Melo et al. [29].

Thirds of the Plant	Average Natural Frequency (Hz)
	${\mathit{\omega }}_{\mathit{n}1}$	${\mathit{\omega }}_{\mathit{n}2}$	${\mathit{\omega }}_{\mathit{n}3}$
Upper	6.45	20.48	43.73
Middle	4.92	15.98	36.15
Lower	5.72	18.12	48.13

.

Melo et al. [17] considered a single geometry, from which the mechanical property data were altered and simulated again. This fact explains how the results of natural frequencies differed from the data of the present study. The natural frequency can also be influenced by the stiffness and mass of the geometry, since the stiffness is directly proportional and the mass is inversely proportional to the natural frequencies. Thus, it is possible to understand that for branch R1, in which all the results were lower than the experimental one, the mass could be decreased, bringing the frequencies obtained via simulation closer to the experimental data. The stiffness module adjustment could also be an option, which could be slightly lower to mitigate this difference.

Although only the first three frequencies were evident when analyzing the experimental results, frequencies related to other vibrational modes could be observed when plotting the raw data, with fewer protruding frequency apices, and were presented in the simulation as belonging to the other modes and observed in computer simulations.

In this sense, the Shapiro–Wilk normality analysis (

Table 10. Test for normality (Shapiro–Wilk).

Experimental	Simulated	W	p
R1	R1	0.888	0.190
R2	R2	0.845	0.109
R3	R3	0.885	0.248

Note: a small p value suggests a violation of the normality assumption.

), the Student’s t test (

Table 11. t test for paired samples.

Experimental	Simulated		Statistic	Degrees of Freedom	p
R1	R1	t Student	0.843	8	0.424
R2	R2	t Student	1.166	6	0.288
R3	R3	t Student	0.822	6	0.443

), and the coefficient of determination R² (

Table 12. Coefficient of determination (R and R²) and linear regression equation measures of fit of the model.

Model	R	R2	Linear Regression Equation
R1	0.988	0.977	f(x) = −6.07x + 1.04
R2	0.981	0.962	f(x) = 1.154x + 0.866
R3	0.974	0.949	(x) = −0.793x + 1.083

) were performed by correlating the data groups for these modes obtained by computer compared to the experimental results for the branches R1, R2, and R3, respectively. These statistical analyses were conducted using the Jamovi software v. 2.6.26 [30], which is based on the R statistical language and environment [31]. Based on these data, it was possible to observe that the calculated F value was lower than that shown in the table; thus, the null hypothesis is accepted, and it can be said that, statistically speaking, there are no significant differences between the groups.

Some present some limitations that must be considered when interpreting the results. The mechanical modeling of the branches was based on the assumption of isotropic behavior using averaged properties obtained from tensile tests, which may not fully capture the anisotropic and heterogeneous nature of lignified plant tissues. Additionally, although the use of 3D scanning enabled high-fidelity geometric modeling, this approach results in complex meshes and higher computational costs, potentially limiting its application to large-scale or real-time simulations. The dynamic analysis was restricted to laboratory conditions, using simplified boundary constraints that may differ from the actual interaction between the branch and the plant during harvesting. Despite these constraints, the correlation between experimental and simulated results demonstrates that this methodology is a promising tool for future development of more accurate, geometry-based computational models, which can support advances in agricultural machinery design.

5. Conclusions

This study aimed to contribute to the understanding of the plagiotropic branches of coffee trees and their dynamic properties using geometries with high fidelity resulting from scanning, which provided interesting results.

The validation of the initial scanning method showed the potential that the proposed methodology can offer. If properly dimensioned, the reality can be reproduced well, in which the simulated average stress obtained a variation of 12.47% of the experimental result and of only 0.02% in the deformation.

The scanning method also proved to be an efficient method for the development of complex geometries since the final geometry showed average deviations of tenths of a millimeter, ensuring high fidelity to the real specimen.

Regarding the studied branches, there were differences between the natural frequencies obtained by simulation and experimentally. R1 obtained percentage differences of −20.022%, −2.19%, and −23.17% for the first, second, and third frequencies, respectively. R2 obtained percentage differences of 7.018%, 7.34%, and −9.88% for the first, second, and third frequencies, respectively. R3 obtained percentage differences of −15.58%, −9.19%, and 2.62% for the first three frequencies, respectively. However, according to the variance tests, despite these differences, there were no significant differences between the frequency data and their experimental and computational modes.

The results obtained in this study represent a significant technological advance that may, in the future, impact the development, improvement, and optimization of coffee harvesters. This work serves as a foundational study for the creation of computational models based on real and organic structures, which capture not only the geometric shape of the branches but also their natural imperfections and singularities. Although this structural complexity may entail increased computational cost, it offers a significant improvement over simplified generic models by providing more realistic and accurate simulations. The ability to faithfully represent the complexity of the branches allows for more precise tuning of vibration parameters such as frequency, amplitude, and duration, tailored to specific mechanical characteristics, which can reduce plant damage, minimize defoliation, and enhance harvesting efficiency. Thus, this work represents the first step toward a new direction in the development of robust and detailed models that may underpin future innovations in more effective and selective agricultural equipment.