J. R. Cooke

A mathematical study of resonance in intact fruits and vegetables using a 3-media elastic sphere model

Abstract A conceptual framework for the interpretation of non-destructive, resonance studies of the texture of intact fruits and vegetables is developed. The internal mechanical properties of intact specimens (e.g. apples, peaches and cantaloupes) modelled as having 3 mechanically distinct, concentric, spherical regions are mathematically related to the intact resonant frequencies and the associated mode shapes are identified. Also, the inverse problem, that of inferring the shear moduli from frequency measurements, is developed and a computer implementation of the computational procedure is described.

Hopf bifurcation in a stomatal oscillator

Stomata are microscopic openings in leaves of green plants which permit gas exchange. This paper presents a parameter study of a model of a stomatal oscillator first derived by Delwiche and Cooke in 1977. We prove the existence of an unstable limit cycle by using the theory of the Hopf bifurcation. Other bifurcations exhibited by the model are also discussed.

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

Abstract The non-linear, normal mode free vibrations of the fruit and stem are analysed as a double pendulum with elastic couplings. The in-phase mode is shown to agree closely with the linear analysis; the out-of-phase mode is shown to be strongly amplitude-dependent. Numerical integration of the coupled non-linear equations of motion indicates that the approximate analytical results are valid for angles as large as one rad. Calculations for apples, cherries, grapefuit, peaches, lemons, oranges, olives, coffee, plums and raspberries are presented and some implications for shaker equipment operation are indicated; e.g. it is conjectured that for fruits to be harvested without stems attached, a pulsating frequency will produce more efficient fruit detachment of the desired type than would continuous shaking.

Dynamics of coupled stomatal oscillators

Stomata are microscopic openings in the leaves of green plants which permit gas exchange. Stomata exhibit oscillatory opening and closing behavior under certain environmental conditions in addition to a daily (diurnal) cycle. In order to explore the effects of coupling between neighboring stomata we present a mathematical model of the dynamics of a system of N coupled stomatal oscillators. An individual stomate is modeled to either remain closed, oscillate periodically, or remain open, depending on the local water potential. Coupling between neighboring stomata is accomplished in the model by taking into account the flow of water in the leaf as well as by oscillator phase coupling.Analysis of the model shows that under certain conditions it exhibits a stable spatially uniform synchronized behavior, referred to here as the in-phase mode. It is also shown that under non-uniform illumination the system may behave in a more complicated fashion.

Role of stomatal oscillations on transpiration, assimilation and water-use efficiency of plants

Abstract The objective of this study was to evaluate the average transpiration rate and CO2 assimilation rate and water-use efficiency during steady-state limit cycle oscillations (time varying cyclic changes) in stomatal pore width both for a hydropassive feedback loop model and for a comprehensive model with both hydropassive and CO2 feedback effects. The results were compared with the corresponding values for the non-time varying pore width (pore width corresponding to a theoretically predicted unstable focus). The hydropassive model predicts that the oscillations of stomata under dry ambient conditions conserve moisture. The comprehensive model which incorporates CO2 feedback loop suggests that such oscillations not only reduce transpiration rate, but also improve water-use efficiency.

A mathematical model of the effects of Co2 on stomatal dynamics

Abstract We extend a previous model of stomatal dynamics (Delwiche & Cooke 1977) which accounted for hydraulic feedback effects but which omitted CO 2 feedback effects. The present work includes both feedback loops in a model featuring the CO 2 chemistry of the guard cell. The model consists of three first order non-linear differential equations which are treated by numerical integration.

Fluid Dynamics of Phloem Flow: An Axisymmetric Model

ABSTRACT FLOW through sieve tubes with sieve plates in the phloem of plants is studied by using an idealized single-pore, axisymmetrical model. The sap is modeled by the steady creeping motion of an incompressible viscous fluid. Numerical results are presented for the as-sociated pressure drop and comparison is made with the simplified resistance formula currently in use in the plant physiology literature.

A finite element analysis of the mechanical and thermal strength of avian eggs

The hot water washing of eggs has been implicated as one source of breakage. A method for estimating shell stresses during and after washing is outlined in this paper. The problem is posed mathematically as a coupled thermo-elastic problem which relates the development of internal pressure to the shrinkage of the shell and the expansion of the inner contents due to heating. Any internal pressure developed in turn stresses the eggshell. In this study we employ a finite element method to determine the effects of various parameters, including: egg size and shape, shell thickness and permeability, wash water temperature and duration of exposure. Results from this study agree closely with previous analytical results for the case of a round egg. More realistic eggshell geometries are considered in this study; the magnitude of stresses are predicted to be larger than those developed in the spherical shell model. The location of maximum stress varied with egg shape; substantial tensile stresses developed after 60 s washing. Internal pressure, and hence shell stresses, diminished rapidly upon removal of the wash water temperature at the shell surface. Incorporation of a static 10N load at the pole in conjunction with thermally-induced stresses greatly increased shell stresses, illustrating that hot water washing, combined with mechanical loads, creates a situation in which more eggs are likely to fail. A cool-down period of 30 s-60 s is predicted to diminish thermal stresses by 60–70%.

A fluid-filled spherical shell model of the thermo-elastic behaviour of avian eggs

Abstract Egg shell breakage during hot water washing results from the differences in the volumetric expansion characteristics of the shell and its contents. Upon heating, the yolk and albumen expand while the enclosed volume of a hollow egg shell has an anomolous shrinkage response. An incompressible liquid and compressible gas-filled elastic, spherical shell model is developed to examine the thermoelastic response to hot water washing. Two bounding cases of a shell which is (1) completely impervious and (2) completely porous to air cell leakage are considered. Egg shell strength is known to decrease with increasing temperature. When combined with internal pressures induced by the washing process, a temporary, but appreciable reduction in the shells capacity to withstand mechanical loads is predicted. Increases in shell porosity, air cell volume and egg shell size decrease the thermally produced mechanical stress. On the other hand, shell thickness does not appreciably influence the thermally induced stress.

Limb Impact Harvesting, Part I: Finite Element Analysis

ABSTRACT A finite element model of the dynamics of limb im-pact harvesting is presented. The model is based upon linearized beam theory and accounts for transverse shear and includes consideration of the leaves, twigs, secondary branches and fruits. The specific fruit of in-terest is modelled as a spherical pendulum. Newmarks direct integration scheme was used to obtain the tran-sient response.