Finite Element Analysis of Modes of Vibration in Apples

Abstract

Three‐dimensional finite element models were developed to study modes of vibration in apples. The apple models were created using two mathematical equations, in which the geometry of the apple was described by a four‐parameter equation and the core structure by an exponential function. Modal analyses were performed to identify the vibrational modes and to study the effects of material properties, structure (i.e., the skin and core), shape, and size on them. There were three classes of vibrational modes in apples: torsional, spheroidal, and nonaxisymmetric. A majority of modes between 0 and 2000 Hz belonged to the nonaxisymmetric class. The square of natural frequency was linearly related to Young's modulus of the apple. The presence of apple skin could increase the natural frequency by 7% for torsional modes and less than 4% for spheroidal modes. As the elastic modulus of the core was doubled, the natural frequency increased by up to 7% for spheroidal modes and well below 1% for torsional modes. The natural frequency for the first mode of each class decreased by no more than 8% as Poisson's ratio was doubled. Apple shape had a large effect on the natural frequencies of all three mode classes; spheroidal modes, however, were less affected by apple shape. The natural frequencies decreased with increasing apple size, but use of the firmness index f2m2/3, where f is the natural frequency and m is the mass of the apple, minimized the size effect.

Characterising pressure and bruising in apple fruit

“…Linear elastic material properties were also assumed, and while this is probably not valid for apple flesh, this is how it has been modelled in previous work [14,15], so as a first attempt it was thought reasonable. A sensitivity analysis could be carried out on the flesh properties to ascertain their influence on the results, but was not done as part of this work.…”

“…To simplify the modelling, isotropic properties were assumed, as they have been in most previous FE studies of apples [14,15]. A Young's modulus of 4MPa was used, as determined for 'Golden Delicious' flesh in the appropriate region of the apple by Abbot & Lu [16].…”

“…Finite Element (FE) techniques have been used previously to investigate modes of vibration in apples [14] and to study transient responses of apples to impulse excitations to determine factors influencing sonic measurements [15]. These are used to ascertain apple firmness, which is a good indication of how ripe an apple is.…”

Modeling of carrot root radial press process for different shapes of loading elements using the finite element method

“…Because of the complex structure of the fruit, the tetrahedral finite element model with four nodes was applied. For comparison, the authors of previous studies [37,39] have determined an experimental model on the basis of isotropic properties of biological material. Abbott and Lu [41] emphasised that often the difficulty in modelling is resulted from differences in mechanical properties of the tested elements (skin, flesh, core).…”

“…The first implementation of the finite element method including apples was conducted by Lu and Abbott [37] and Puri and Anantheswaran [38] . The authors analysed a transfer of vibrations by the fruit tissue while the apple impacts against another element or fruit.…”