A Review of Analytical Methods for Solid Rocket Motor Grain Analysis

Analytical methods for solid rocket motor grain design are proving to be tremendously beneficial to some recent efforts to optimize solid-rocket propelled missiles. The analytical approach has fallen out of favor in recent decades; however, for some classes of grains, the analytical methods are much more efficient than grid-based techniques. This paper provides a review of analytical methods for calculating burn area and port area for a variety of cylindrically perforated solid rocket motor grains. The equations for the star, long spoke wagon wheel, and dendrite grains are summarized and the development of the burn-back equations for the short spoke wagon wheel and the truncated star configurations are included. This set of geometries and combinations of these geometries represent a very wide range of possibilities for two-dimensional grain design. Introduction In many practical solid rocket motor design efforts, final geometric designs for grains are arrived at using numerical layering techniques. This process is geometrically versatile and imminently practical for cases in which small numbers of final geometries are to be considered. However, for a grain design optimization process in which large numbers of grain configurations are to be considered, generating grids for each candidate design is often prohibitive. For such optimization processes, analytical developments of burn perimeter and port area for two-dimensional grains are critically important. Most modern texts on solid rocket propulsion do not provide geometric details for grain design. This paper will offer a review of analytical methods for determining burn area and port area as a function of burn distance for a selection of common grain designs. Analytical developments for solid rocket motor grains were much more prevalent in the decades before widespread use of microcomputers. A summary of one version of the burn back equations for the star grain and for part of one type of wagon wheel can be found in Barrere. Analytical methods have also been developed for the truncated star and for the dendrite. Other potential grain configurations are described in NASA publications but very few geometric details are given in such publications. 1 39th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit 20-23 July 2003, Huntsville, Alabama AIAA 2003-4506 Copyright © 2003 by Roy Hartfield. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Star Grain Geometry Grain geometries are generally described using lengths and angles defined in the cross section. A sample burn back diagram is shown in Figure 1 and the geometric definition diagram for the familiar star grain geometry is shown in Figure 2. Since some of the geometric equations for the burn perimeter and port area for the star grain have been developed and published, only a summary of this geometry is included here. Figure.1 Sample Star Grain Figure 2. Star Grain Geometry The geometric relationship between the two primary angles in the geometry definition can be written as: