Finite element analysis of vibration mode and firmness evaluation for Korla pear

The firmness of fruits can be estimated with measurement of its natural frequency (ƒ). Considering the effects of mass (m) and density (ρ) on natural frequency of fruits, the different firmness indexes ƒ2m, ƒ2m2/3 and ƒ2m2/3ρ1/3 were proposed. Nevertheless, some studies showed that the natural frequency of fruit was also affected by fruit shape. Consequently, the calculated values of these indexes (e.g. ƒ2m2/3ρ1/3) were inaccurate for firmness evaluation of fruit. However, little information is reported on the firmness index considering the fruit shape. Because the finite element modal analysis has been widely used to obtain natural frequency and mode shapes of fruits and vegetables, so in this work it was applied to provide data of natural frequencies of Korla pear with different shapes and the new firmness index including the fruit shape factor was then established. The roundness ratio (q) of longitudinal cross section was employed to describe the fruit shape of Korla pear quantitatively. It is expressed as q=a/b, where a, b are the major axis (longest intercept) and intermediate axis (longest intercept normal to a) of longitudinal cross section. The previous index ƒ2m2/3ρ1/3 was transformed into ƒ2m2/3ρ1/3qt introducing the power function of roundness ratio qt. The outline of a real Korla pear was determined using the “Spline” option and the 3D solid model was created by rotating the half contour plane around the major axis. Assuming Korla pear as linear elastic and isotropic material, the model was free meshed using 3D tetrahedral structural solid element with 10 nodes (SOLID 187) in ANSYS Workbench 14.0. The first 20 natural frequencies and mode shapes were extracted from the free modal analysis results. In this work, range ratio (R.R) and coefficient of variation (C.V) were used to analyze the variation of the calculated values of the firmness index. The optimal values of t for different vibration modes, which were corresponding to the maximum of range ratio, were calculated through polynomial fitting the data of the range ratio. Large excitation energy could cause the Korla pears to be damaged, so the first three vibration modes with relative lower natural frequency were selected for analysis, namely torsion mode, bending mode and oblate-prolate mode. The largest deformation appeared at the stem end and the calyx end for selected mode shapes in the finite element modal analysis, so both ends of the Korla pear were suitable positions to excite and detect vibration signal. The results of ANOVA showed that the roundness ratio (q) describing Korla pear shape had significant effect on natural frequencies of these modes. For the new firmness index ƒ2m2/3ρ1/3qt, the optimal values of t for different vibration modes were 1.19 for torsion mode, 1.47 for bending mode and 0.47 for oblate-prolate mode. Compared with index ƒ2m2/3ρ1/3, the range ratio of index ƒ2m2/3ρ1/3qt was larger and the coefficient of variation was smaller obviously. Taking bending mode for example, the value of range ratio increased from 55.08% to 96.18%, whereas the coefficient of variation reduces from 21.97% to 0.61%. As a result, it indicated that the index ƒ2m2/3ρ1/3qt could replace with the index ƒ2m2/3ρ1/3 to accurately evaluate the firmness of Korla pear. Furthermore, this firmness index can provide a theoretical basis and practical guidance for vibration detection of internal quality of Korla pear