A universal throw model and its applications

Abstract

A deterministic model has been developed that describes the throw of debris or fragments from a source with an arbitrary geometry and for arbitrary initial conditions. The initial conditions are defined by the distributions of mass, launch velocity and launch direction. The item density in an exposed area, i.e. the number of impacting debris or fragments per unit of area, has been expressed analytically in terms of these initial conditions. While existing models make use of the Monte Carlo technique, the present model uses the source function theorem, an underlying mathematical relation between the debris density and the initial distributions. This gives fundamental insight in the phenomenon of throw, and dramatically reduces the required number of trajectory calculations. The model has been formulated for four basic source geometries: a point source, a vertical cylinder, a horizontal cylinder, and a vertical plane. In combination with trajectory calculations the item density can be quantified. As an illustration of the model, analytical results are presented and compared for the vertical plane and the vertical cylinder geometry under simplified assumptions. If uncertainties exist in the initial conditions, the model can be used to investigate these initial conditions based on experimental data. This has been illustrated on the basis of a trial with 5 ton of ammunition stacked in an ISO container. In this case the model has been successfully applied to determine the debris launch angle and velocity distribution, by means of backward calculations. If, on the other hand, sufficient information on the initial conditions is available, the model can be used as an effect model in risk assessment methods, or for the requirements on protective measures. The model can be used to predict safety distances based on any desired criterion.