Empirical formula for the natural frequency of Camellia oleifera trunk using allometric growth

Camellia oleifera can significantly guarantee food safety in the edible oil industry. The manual harvesting of Camellia oleifera has greatly restricted the full mechanization of the Camellia oleifera industry. Vibratory mechanized harvesting has been adopted with the high harvesting efficiency of Camellia oleifera fruit. Existing studies have also proved that the vibratory picking frequency can directly dominate the detachment of forest fruits. There is a high demand to establish the dynamics model of the Camellia oleifera tree, in order to calculate the trunk's natural frequency. In this study, an ideal fractal tree model was investigated for the Camellia oleifera, according to the sympodial branching and allometric growth. The bifurcated basic unit was extracted from the fractal tree model. Three dynamics models were then deduced with the point masses at different locations using the classic mass-spring analysis. Consequently, the theoretical expressions of the first-order natural frequency were derived for three dynamics models. Taking Camellia oleifera (variety: Huaxin) in Hunan Province as an application example, the lateral branching ratio and the slenderness coefficient were identified to measure the morphological parameters of the tree. Subsequently, the branch density and flexural elastic modulus were measured by water immersion and a three-point bending test. The theoretical values of the first-order natural frequency were calculated for the bifurcated basic unit with different branching angles, in order to substitute the identified tree parameters. Then, a finite element simulation model of the bifurcated basic unit for Camellia oleifera was constructed to simulate the natural frequency. Compared with the simulated and theoretical frequency value, the dynamics model with point mass equally distributed at both rod ends was the closest to the simulated model, with an average error of 7.3%. Finally, the mathematical formula was obtained with the first-order natural frequency of the cantilever beam with point mass at the rod top. An empirical calculation formula was derived for the bifurcated basic unit using parameter identification. The smallest error was found between the theoretical natural frequency from the empirical formula and the simulated value, with a maximum error of 0.41%. The validity and accuracy of the empirical formula were verified. The finding can provide the theoretical guidelines to optimize the vibration parameters of fruit harvesting machines for Camellia oleifera.