852. A Surface Integral Method for Computer Calculation of Mass Properties

Paper

Abstract

A computer technique is described for the calculation of weights, moments, and centers of mass of general solids, the surfaces of which are defined in an incremental or digitized manner. The method is based on surface integral formulations and has the advantage of eliminating the need to decompose a given solid into standard geometrical forms. Experience with this technique over the past two years has proven it to be efficient and practical in terms of both computer usage and input-output considerations. The basic computation routine is particularly simple and could be adapted for use on small desk-top computers, although experience has been limited to Fortran implementation on an IBM 360-40 machine. The mathematical formulation is derived from the familiar volume integral definitions through application of Gauss’ divergence theorem. Typical of this transformation is the moment equation about the x-axis in rectangular Cartesian coordinates given below. Details are presented for the application of these equations to two classes of solids: solids of revolution and general polyhedrons. Since any solid can be approximated by a polyhedron, this latter approach, theoretically, could be applied to all problems. However, practical input considerations often preclude such an approach unless a digitized surface description is available. In the solid-of-revolution program, direct solutions to the integral equations are presented for a straight-line element description of the cross section of the solid. This has proven to be the simplest input format encountered, and it is one that can be obtained directly from a mechanical digitizer. In the general polyhedron program, numerical integration techniques based on Gauss-Radau coefficients are employed. It has proved advantageous to obtain computer plots of the program outputs for checking results of individual components as well as complete assemblies. Examples of these plots and practical considerations of their use are discussed. Section views are shown for axisymmetric components, together with perspective projections of general polyhedrons with hidden lines eliminated. Experience with these programs to date is discussed and computer requirements, average run times, accuracies, and overall computational efficiencies are described.