386. A Graphical Technique for Calculating Time-Changing Moments of Inertia

Paper

Abstract

This paper was presented at the Twenty-second Annual National Conference of the Society of Aeronautical Weight Engineers at St. Louis, Missouri, April 29-May 1, 1963. In this age of complex missiles and rapid technological advances the weight man finds himself continually facing new and more complex problems. The Emerson Electric Weight Department was recently faced with such a problem.
The problem involved the weight moment of inertia of a propellant of complex design at any instant of burning during a period ranging from ignition to burn-out. To solve this problem we evaluated different techniques of approach in order to choose the one which combined accuracy with flexibility and low complexity. The techniques evaluated include curve filling and special formulae, extended use of standard formulae, scaling curves with the polar compensating planimeter, and the ‘graphical summator’.
The results of the evaluation proved the ‘graphical summator’ to be the only technique that combined high accuracy and flexibility with only a moderate degree of complexity. For this reason the ‘graphical summator’ was the technique utilized for the calculations.
This graphical technique exploits the basic definition of areas and moments by the use of finite summations as accurate approximations to infinitesimal summations, or integrations. The ‘graphical summator’ is a polar coordinate grid in which the radial lines have a fixed angular spacing. Each polar cell will have an area and inertia that depends only upon its radial distance from the pole. A weighting factor for inertia is determined for each radial increment between polar cells. In actual use the summator is placed over the desired shape with the pole of the summator coincident with the axis of rotation. The cells falling under the desired curves are summed allowing for partial occultation of the cells. This sum is multiplied by the appropriate weighting factor and the inertia obtained.
This technique can be adapted to many tasks requiring the calculation of area, volume, weight or moment of inertia. Only after weight engineers throughout the industry utilize, adapt and improve on the ‘graphical summator’ technique will its full potential is realized.