Abstract
Apple fruit is subjected to operations such as harvesting, packaging, grading, storage, and transportation. During these stages, numerous static and dynamic loads are applied to individual fruits, leading to mechanical damage, quality deterioration, and consequently a reduction in marketability and export potential. The objective of this study was to analyze the dynamic behavior of export apples of the Golden Delicious and Red Delicious cultivars against impact-induced damage using finite element simulation. In addition, using an Instron texture analyzer available in the rheology laboratory, the mechanical properties of the apple peel, flesh, and core were determined separately. The measured physical and mechanical properties were then used as input data in the engineering data section of Abaqus software. To achieve the objectives of the study, an impact test was first conducted in a separate section as a factorial experiment with independent variables including two apple cultivars, three impact energy levels (0.0124, 0.0238, and 0.0712 J), and two curvature radii of the impact location (38.8 and 42.2 mm). The results of this study indicated that, in the simulation of the apple impact test using the finite element method, there was a high correlation and reliability between experimental results and modal analysis, such that the coefficient of determination (R²) obtained from the comparison of experimental and modal frequencies was 0.98 and 0.99 for Golden Delicious and Red Delicious apples, respectively. The results of this study can provide a significant advantage in the design of harvesting and post-harvesting machinery for apples at a commercial scale.

Introduction
Apples are affected by operations such as harvesting, packaging, grading, storage, and transportation on their way from the orchard to the sales centers. During these stages, various static and dynamic loads are applied to each product, which leads to damage and also causes a decrease in quality, and as a result, a decrease in marketability and export of the product. This research aims to analyze the dynamic behavior of export apples of Golden Delicious and Red Delicious varieties against damage caused by impact using finite element simulation, which is defined by defining a viscoelastic material model in the two parts of the flesh and core (central part) of the apple and an elastic-plastic material model in the apple skin section using finite element modal analysis using Abaqus software. In order to investigate the linear regression model of the dynamic behavior of apple fruit, an impact was applied to the apple fruit using three-axis force gauge and accelerometer sensors. The physical properties of the apple fruit were measured in the laboratory. Also, using the Instron texture analyzer available in the rheology laboratory, the mechanical properties of the skin, flesh, and seed of apple fruit were determined separately, and the results of the physical and mechanical properties were used as input in the engineering data section of the Abaqus software. To achieve the objectives of the study, first, in a separate section, an impact test was performed in the form of a factorial experiment with independent variables including two apple varieties, three levels of impact energy (0.0124, 0.0238, and 0.0712 Joules) and two levels of the radius of curvature of the impact site (38.8 and 42.2 mm). The results of this study showed that in simulating the apple impact test using the finite element method, there is a high correlation and reliability between the experimental results and modal analysis, so that the coefficient of determination (R2) resulting from the comparison of experimental frequencies and modal analysis for the Golden Delicious and Red Delicious apple varieties is 0.98 and 0.99, respectively. So that the results of this study can provide a significant advantage for the design of harvesting and post-harvest machines on a commercial scale for apple fruit.
Material and Methods 
To conduct this research, healthy, undamaged export apples of the Golden Delicious and Red Delicious varieties were selected from orchards in Mahabad city (longitude 34.5°45'47" and latitude 9.7°51'36"), the hub of apple production and export in the country, and were harvested manually. The harvested product was transferred to the rheology laboratory of the Faculty of Agriculture, Bu Ali Sina University, in special containers under standard conditions. The fruits were then stored in a refrigerator at 4 degrees Celsius. Before conducting the experiments, the apple fruits were placed at the laboratory temperature to become isothermal with the ambient temperature. Then, mechanical tests and impact tests were performed on the samples.
Results and Discussion  
To determine the natural frequencies of apple fruit by impact testing, the output data from the excited samples were recorded by a four-channel signal processing device (ECON, AVANT Lite, model: MI-6004), and the results were then transferred from the time domain to the frequency domain using the fast Fourier transform in MATLAB software. In the polar direction, the first vibration mode for the Golden Delicious cultivar was sensed at a natural frequency of 125.27 Hz, and the second and third modes had natural frequencies of 151.12 and 185.17 Hz, respectively (Table 4). In the Red Delicious cultivar, the first vibration mode was sensed at a natural frequency of 121.61 Hz, and the second and third modes had natural frequencies of 148.21 and 174.23 Hz, respectively, in the polar direction (Table 6). The first modal frequency (simulated) in the polar direction in the Golden Delicious cultivar was considered to be 138.18 Hz (first mode), the second modal frequency was considered to be 151.45 Hz (second mode), and the third modal frequency was considered to be 204.14 Hz (third mode) (Table 4). Also, in the Red Delicious variety, the first modal frequency in the polar direction was 132.18 Hz (first mode), the second was 155.61 Hz (second mode), and the third was 197.30 Hz (third mode) in the Abaqus simulation (Table 6). In the equatorial direction, the first vibration mode for the Golden Delicious variety was sensed at a natural frequency of 135.52 Hz, and the second and third modes had natural frequencies of 141.16 Hz and 175.17 Hz, respectively (Table 5). In the Red Delicious variety, the first vibration mode was sensed at a natural frequency of 124.18 Hz, and the second and third modes had natural frequencies of 158.25 Hz and 214.23 Hz, respectively, in the equatorial direction (Table 7). The first modal frequency in the equatorial direction in the Golden Delicious cultivar was 141.38 Hz (first mode), the second modal frequency was 155.86 Hz (second mode), and the third modal frequency was 187.45 Hz (third mode) (Table 5). For the Red Delicious cultivar, the first modal frequency in the equatorial direction was 137.16 Hz (first mode), the second modal frequency was 168.42 Hz (second mode), and the third modal frequency was 227.22 Hz (third mode) in the Abaqus software simulation (Table 7). As shown, the differences between these frequencies for both apple fruit groups are highly reliable, so that the linear regression model and the coefficient of determination R2 resulting from the comparison between the experimental and modal frequencies for the Golden Delicious and Red Delicious varieties are 0.98 and 0.99, respectively. These results can provide a significant advantage in the design of commercial-scale harvesting and post-harvest machines for apple fruit.
Conclusions 
In the industry, before manufacturing engineering parts, modal tests are used to determine the material, selecting a more compatible material for parts with greater displacement to extend their life and improve function. Considering this type of application of modal analysis, this approach can be used to genetically modify agricultural products, especially fruits, to change their texture, thereby increasing their useful life. Also, the finite element model is a fast, cost-effective, and non-destructive method for evaluating natural frequencies, especially the resonant frequency, and for providing a regression model with acceptable accuracy for the design of harvesting and post-harvest fruit machines.
Acknowledgements
Bu-Ali Sina University is appreciated for supporting this research in terms of supplying laboratory materials and equipment.
Data Availability Statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Ethical Considerations
Conflict of Interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper or any conflict of interest.
 Ethical Review: This study does not involve any human or animal testing. Informed Consent: Not require
Conflict of Interest
There are no contributions of interest in this article, and this has been confirmed by all authors.
Funding Statement
The author(s) received no specific funding for this research.