INTRODUCTION
The geographical location gives Venezuela potential for tropical agriculture, being able to obtain high yields in permanent crops such as coffee, cocoa, oil palm and fruit trees, or semi-permanent ones such as sugarcane, banana, plantain and cassava (^{FAO, 2009}).
The cultivation of plantain (Musa paradisíaca) and banana (Musa cavendish) in Venezuela represents the largest fruit activity in the country with 459,000 tons in 2013. The production is distributed in different parts of the national territory in relation to the adaptive degree of the cultivar (^{Martínez, 2009a}). The areas with the highest banana production are in the states of Aragua, Carabobo, Trujillo, Mérida, Barinas and Yaracuy. Figures from the agricultural census (May 2007/April 2008) indicate that the State of Merida is the largest producer, followed by Trujillo with a production of 144,000 tons (^{Martínez, 2009b}).
The main banana zone of Trujillo State is characterized by monocultures for export purposes. It is located in the alluvial plain of Motatán River, that, according to figures from the Agricultural Census, represents 15, 75% of the production, meanwhile 84, 25% corresponds to the middle and upper zone of the state, where by tradition there is an agro-ecological production of bananas, generally associated with coffee and citrus category. The main producing municipalities are Monte Carmelo, Boconó, Trujillo, Escuque and San Rafael de Carvajal. ^{Ramírez et al., (2010)} and ^{Lescot, (2013)} point out that high-zone bananas have good quality characteristics for commercialization, but with the condition of poor agronomic management and in the transfer during their harvest, in plantations which cause non appropriate aesthetic quality of the fruit.
Bananas grown under the modality of employer and family farm are harvested manually by one or two people, in order to prevent clusters from falling to the ground and spoiling them, damaging the price and the final quality of the product. In the study area, banana plantations are established at an average altitude of 1,300 meters above sea level and in areas with high slopes (^{IGVSB, 1999}), which greatly hinders transfer work in the banana harvest. The harvest is one of the last operations of the cultivation of bananas and at the same time a key point to obtain the fruit of the desired quality in the market. Carriage of banana clusters from the crop to the point where it is prepared for transportation to the markets, is usually done manually (on the shoulder) and in animals (mules and donkeys), according to ^{Magalhaes et al., (2004)}. In addition, the manual transport of banana clusters, within the plots, causes physical damage to the fruits that compromise their quality. These damages are generally caused by falling and crushing.
Not all countries have systems for handling and transporting freshly harvested agricultural products, from planting to packing plants or shipping. The cable track system (aerial cable transport) of the plantation is as important as the irrigation and drainage system of the plantation, since the transport and quality of the fruit depend on it. This method is undoubtedly the most efficient and economical way of transporting fruit in banana plantations that has been developed in recent years (^{Soto, 2008}).
On the other hand, during investigations related to the working components of agricultural machines in general and in particular in the case of the transport of banana bunches by cable, the determination of the physical-mechanical properties of the agricultural products to be transported results indispensable stage (^{Parra et al., 2006} and ^{2007} and ^{Valdés et al., 2008}). That is due to they constitute input data during the evaluation of the theoretical models (^{Valdés et al., 2018}; ^{Valdés et al., 2012}), elaborated for the calculation of the different design and operating parameters of those machines.
Investigations of these properties related to the object of study have been carried out by ^{Martinez and Mollineda, (2003)}; ^{Velásquez et al. (2005}, ^{2012)}; ^{Ciro et al., (2005)}; ^{Millán y Ciro, (2012)} and ^{Martínez and Bermúdez, (2016)}, but aimed at plantain and bananas directly, not in the case of bunches.
Taking these aspects into account, the objective is to: Determine the physical-mechanical properties of banana clusters (Musa cavendish L.) as object of aerial cable transportation, which are required as input data in theoretical models, in order to design a transport system for banana bunches in the mountainous areas of Trujillo State in Venezuela.
METHODS
Characterization of the Experimental Zone
The area under study is located in the Alto de la Cruz Community, Carvajal Parish of San Rafael de Carvajal Municipality at Trujillo State, in Venezuela. The samples were obtained from four banana harvests of Cavendish variety corresponding to the months of November 2012, January 2013, February 2013 and April 2015. In the four harvests, 36, 22, 26 and 14 bunches were obtained, respectively, for a total of 98 bunches harvested. The relief of the soils is mountainous with altitudes ranging from 800 to 1,400 meters above sea level and with an average slope of 53, 84%, which is equivalent to an inclination angle of 24^{o}. The farm is generally made up of two plots (A and B), which occupy an area of 2,5 and 1,30 ha, respectively, as it is seen in the plain-metric topographic survey (Figure 1a). The function of the system is to transport the load in the indicated path from the lowest part (point 1) to the access road (point 2) where the product is shipped (Figure 1b).
Materials and Methods for Determining Dimensional Characteristics
Rachis diameter (dr). Assuming that the geometry of the rachis cross-section is approximately circular, the diameter of the rachis was calculated by measuring the perimeter (pr) in the first ring of the spine with a measuring tape of 100 cm in length and an accuracy of 1 mm as indicated in Figure 2a. The diameter (dr) is calculated from the following expression:
Longitude of the bunches (Lr). It was measured with the use of a 100 cm long measuring tape with a precision of 1mm, from the last cluster or hand to the first spine ring, as illustrated in Figure 2b.
Bunches diameter (Dr). The perimeter (Pr) was measured in three sections of the cluster (upper, central and lower part), with a measuring tape of 100 cm in length and precision of 1 mm, according to Figure 2c. The total arithmetic mean was determined for the three sections studied. The diameter calculation was performed similar to the rachis diameter.
Materials and Methods for the Determination of Inertial Properties
Bunches Mass (m_{r}). It was determined by weighing the clusters with a digital scale with a capacity of 40 kg and an accuracy of 0,01 kg, as illustrated in Figure 3.
Mass center. The cluster was suspended on a rigid steel hook and moved until its horizontality and verticality were achieved by checking it from a spirit level and two wooden guides, as shown in Figure 4. In this position the coordinates Xc and Yc were measured with respect to the point O located at the left end of the cluster (Y axis) and with respect to the lower part of the rachis (X axis), respectively, guaranteeing the required verticality and horizontality on each axis, with the spirit level. The crossing of these coordinates determines the center of mass of the cluster.
Moment of inertia. The determination of the moment of inertia of the clusters is made with respect to the central axis of the rachis (Figure 5a). It is hung vertically on a hook supported by an elastic steel wire of 0,4 mm in diameter and 0, 5 m in length, forming a torsional pendulum, according to the methodology proposed by ^{Martinez et al. (2006)} and ^{Valdés et al. (2009)}. The period of free torsional oscillations is determined by rotating the cluster-wire system 45^{0} with respect to its equilibrium position and allowing it to oscillate freely, timing the time corresponding to 10 oscillations with a precision level of 0,1 sec. The elastic constant of the steel wire is determined by replacing the cluster in the torsional pendulum with a metal cylinder (Figure 5b) whose moment of inertia is determined by the expression:
The frequency fc of the free oscillations of the torsional pendulum formed by the metal cylinder and the steel wire is related to the moment of inertia of the cylinder by the expression:
Where: K-torsional elastic wire constant; Nm/rad;
By subjecting the cylinder to free oscillations and measuring the time corresponding to 10 oscillations, the frequency fc is determined as the inverse of the oscillation period, and then the elastic constant K of the wire can be determined from the expression (3).
Once the elastic constant of the wire is known, the moment of inertia I_{y} of the cluster is determined by the expression:
where: fr- frequency of free oscillations of the torsional pendulum formed by the cluster - wire system, Hz
Statistical Processing of Primary Data. Four samples of clusters were taken in which the following parameters were measured: rachis perimeter, length, perimeter, mass and inertia of the clusters. A descriptive analysis of all samples was performed independently, with the determination of the measures of central tendency (arithmetic mean) and dispersion measures (standard deviation and coefficient of variation). In addition, an analysis of variance of LSD type was performed, where the p-value among the four harvests was determined to identify if there were significant differences between the means.
On the other hand, the simple and multiple regression method was applied to determine the dependence of the moment of inertia as a function of mass and diameter. Finally, the t-test was performed determining the level of significance in the single and multiple regression model. For the data processing and analysis, Microsoft Excel version 2013 and Statgraphics version 16 software were used.
RESULTS AND DISCUSSION
Results of the Determination of the Dimensional Characteristics
Table 1 shows the results of the descriptive analysis performed on the rachis diameter, length and diameter of the clusters. The arithmetic mean obtained from the rachis diameter for the 4 harvests mounted to 5,40 cm, with a standard deviation of ± 0,77 cm and a coefficient of variation of 14,25%, hence it is inferred that the dispersion of the data is considered adequate since it does not exceed 15% variation with respect to the average. The simple variance analysis performed when comparing the four crops indicated that there are no significant differences between the rachis diameter averages, since the p-value is 0,6123 greater than 0,05, this further explains the homogeneity of this parameter in the harvests made.
Statigraph | dr | Lr | Dr |
---|---|---|---|
Arithmetic mean, cm | 5,40 | 63,96 | 33,67 |
Standard deviation, cm | 0,77 | 14,92 | 2,69 |
Coefficient of variation, % | 14,25 | 23,33 | 7,99 |
The arithmetic mean obtained in the diameter of the clusters for the 4 crops amounted to 33,67 cm, with a standard deviation of ± 2,69 cm and a coefficient of variation of 7,99%, which allowed considering that the dispersion of data is adequate since it does not exceed 15% variation with respect to the average.
The analysis of simple variance performed when comparing the diameter of the four harvests, indicated that there are no statistically significant differences between the means of the cluster diameter, since the p-value is 0,2182 greater than 0,05. That explains, in addition, the homogeneity of this parameter in the harvests made.
The arithmetic mean obtained for the length of the clusters, in the 4 harvests, amounted to 63,96 cm, with a standard deviation of ± 14,92 cm and a coefficient of variation of 23,33%, which indicates a slight dispersion because it exceeds the range of 15-20% variation with respect to the average. The comparison of the clusters length for the four crops by means of a simple analysis of variance indicated that there are statistically significant differences between the means of the length, since the p-value is 0,0157 less than 0,05 for a 95 % confidence. The application of a multi-range test showed that the length of the clusters in the fourth harvest differs from the first three as seen in Table 2. This variability may be given by the climatic conditions of the area, since the crop is produced in the dry land modality and approximately six months pass, before the harvest of the fruits is made. Consequently, the fruit formation is influenced by adverse climatic variations such as high temperatures and drought, causing irregular sizes of banana clusters, especially due to the low accumulated rainfall in April, the time when it is harvested.
Results of the Determination of Inertial Properties
Table 3 shows the results of the descriptive analysis performed on the mass, the coordinates (X, Y) of the center of mass and moment of inertia of the clusters. The arithmetic average obtained in the mass of the clusters for the four crops amounted to 16,58 kg, with a standard deviation of ± 4,91 kg and a coefficient of variation of 29,65%, which denotes a slight dispersion of the data since it exceeds 15-20% of variation with respect to the average.
The arithmetic mean obtained in the Xc coordinate of the center of mass for the fourth harvest amounted to 16,73 cm, with a standard deviation of ± 1,37 cm and a coefficient of variation of 8,20%, therefore, it is considered that the dispersion of the data is adequate since it does not exceed 15% of variation with respect to the average.
The arithmetic mean obtained in the Yc coordinate of the center of mass for the fourth harvest amounted to 28,84 cm, with a standard deviation of ± 3,78 cm and a coefficient of variation of 13,11%, therefore, it is considered that the dispersion of the data is adequate since it does not exceed 15% of variation with respect to the average.
Statistic Parameter | Mr, kg | Xc, cm | Yc, cm | I_{y}, Kg m^{2} |
---|---|---|---|---|
Arithmetic mean | 16,58 | 16,73 | 28,84 | 0,13 |
Standard deviation | 4,91 | 1,37 | 3,78 | 0,11 |
Coefficient of variation, % | 29,65 | 8,20 | 13,11 | 79,39 |
The arithmetic mean of the moment of inertia of the clusters obtained, for the fourth harvest amounted to 0,13 kg m^{2} with a standard deviation of ± 0,11 kg m^{2} and a variation coefficient of 79,39%. That indicates a high dispersion of the data since it exceeds 15-20% variation with respect to the average, this was due to the adverse weather conditions mentioned above.
The simple variance analysis carried out when comparing the mass in the four crops indicated that there are statistically significant differences between the means of the clusters' masses, since the p-value is 0,000 less than 0,05, for a confidence level of 95%. The application of a multi-range test showed that the mass of the clusters in the fourth harvest differs from the first three as seen in Table 4, this is a product of the climatic variability mentioned above.
Relation of the Moment of Inertia Depending on the Mass of the Clusters
The regression analysis showed that the Box-Cox Method performs the best adjustment of both variables, with a correlation coefficient of 93%, a determination coefficient (R^{2}) with value of 86,65% greater than 75% and a standard error of 0,023 (Figure 6). These results indicate that there is a strong functional dependence between both variables. This method performs transformations in the primary data so that the quadratic error is guaranteed to be minimal, from the following linear equation.
The following transformed equation is obtained:
Relation of the Moment of Inertia Depending on the Diameter of the Bunches
The regression analysis performed between the mentioned variables showed that the polynomial model makes the best fit, obtaining a correlation value of 1,37, standard error of 0,11 and an R^{2} of 88,77%, as shown in Figure 7. In addition, a p-value of 0,012 less than 0,05 was obtained, which indicates that there is a strong functional dependence and a statistically significant relationship between the moment of inertia and the diameter of the clusters with a confidence level of 95 %, obtaining the following equation:
CONCLUSIONS
As a result of the investigation, the values of the physical-mechanical properties (dimensional and inertial) of the Cavendish variety of banana clusters were obtained. They are required as input data to the theoretical models that make it possible to design a cable transport system for clusters in the mountainous area at Trujillo State, in Venezuela. The main results were the following:
Dimensional Characteristics
The average diameter of the rachis for the four crops was 5,40 cm with a coefficient of variation of 14,25%, while the average diameter of the clusters for the four crops was 33,67 cm with a coefficient of variation of 7,99%, evidencing in both cases a low variability of the data obtained.
Regarding the average length of the clusters for the four crops, the average value reached 63,6 cm with a coefficient of variation of 23,33%, showing a slight dispersion of the data obtained.
Inertial Properties
The average mass of the clusters for the four crops was 16,58 kg with a coefficient of variation of 29,65%, showing a slight dispersion of the data obtained;
The average values of the coordinates of the center of mass were Xc = 16,73 cm and Yc = 28,84 cm, with coefficients of variation of 8,20% and 13,11%, respectively, hence, the dispersion of the data is considered adequate;
The average moment of inertia obtained was 0,13 kg m^{2} and as a result of a regression analysis that related this property to the mass through the Box-Cox Method, an R^{2} = 86,65% was obtained. Regarding the relationship of this property with the diameter of the clusters, the best fit model was the Polynomial with a value of R^{2} = 88,7%. That indicates, in both cases, a strong dependence between the related variables. These models can be used to predict the moment of inertia.