Bispherical Coordinate System Research Articles

Vibratory fruit harvesting: A linear theory of fruit-stem dynamics

A linearized, three-degree-of-freedom model of a fruit-stem system which is excited by simultaneous, periodic horizontal and vertical planar displacements (any straight line, ellipse, or circle) of the supporting structure was examined in relation to vibratory fruit harvesting. The regions of dynamic instability and the corresponding modal shapes were inferred from the coupled, nonhomogeneous Hill's equations which describe the motion of the double physical pendulum with torsion springs. The frequency conditions for fruit separation or detachment with or without the stem attached to the fruit may be predicted. Although the model is generally applicable to all freely, vertically suspended fruits, special consideration was given to apples. The surface area, volume, and moments of inertia for the apple, orange, and lemon-shaped geometries of a bispherical coordinate system were analytically determined. References to the available data on the relevant physical properties of fruits and stems were given.

Stress Distributions Analyzed in Bispherical Co-Ordinates

Boussinesq-Papkovich potentials are used in conjunction with the bispherical co-ordinate system to analyze three problems in the classical theory of linear elasticity: (a) The extension of the Boussinesq point-load problem to that in which the half-space contains a spherical cavity; (b) the determination of the stress distribution in an eccentric spherical shell under uniform internal pressure; (c) the determination of the stress distribution in a half-space containing a uniformly pressurized spherical cavity. Numerical results are presented for representative configurations and load distributions in each case.