A Firmness Index for Fruits of Ellipsoidal Shape

This paper proposes a novel fruit firmness index c3 for solid ellipsoids with one long principal axis and two equal short principal axes. The firmness index c3 is dependent on the mass, density, and two natural frequencies of the lowest spherical modes (under free boundary conditions) moving along long and short axes, respectively. In fact, the new index c3 is a generalisation of the conventional firmness index for a sphere, because the two natural frequencies for an ellipsoid become equal for a sphere. This newly developed index extends the firmness estimation for fruits or vegetables from a spherical to an ellipsoidal shape. The formula was developed according to the following steps: (1) assign mode shape functions observed from a finite element model; (2) approximate the potential energy and kinetic energy over the whole volume; (3) apply Rayleighʹs quotient to determine natural frequencies in a simple form; and (4) deduce an appropriate mathematical equation relating firmness (equivalent to modulus of elasticity) with natural frequencies, mass, and density. Detailed derivations are provided, together with finite element models for verifications of the new firmness index. Satisfactory results from simulations show that the proposed formula is quite suitable for predicting the firmness of ellipsoidal fruits or vegetables.