Surface Accuracy Analysis of a Regular Polygonal Membrane in a Space Reflector

Abstract

The surface accuracy of a N-sided regular polygonal membrane in a reflector is investigated analytically, where N is an integer of 3, 4, 5, 6, ---. The membrane is assumed to be located at an arbitrary point on the reflector surface with an arbitrary rotation angle between the side of the membrane and the direction of the principal curvature of the ideal parabolic surface. Membranes with two kinds of edge deflections are considered. One is a coincident-edge membrane whose edges coincide with the approximate parabolic surface of the local point where the membrane is placed. The second is a coincident-edge-optimum membrane, which is obtained by minimizing the rms error with the normal translation of the coincident-edge membrane. Analytical formulation for the deflection and rms error is obtained based on a linear membrane equation. Effects of edge deflections, number of sides N, location of the membrane, etc. upon the surface accuracy are examined. The surface error of the coincident-edge-optimum membrane is found to be about half that of the coincident-edge membrane. The surface error of each membrane decreases as the distance from the center axis of the ideal parabolic surface becomes larger and is not dependent on the rotation angle. The surface errors of coincident-edge and coincident-edge-optimum membranes increase monotonically as N increases. These surface errors converge to those of a circular membrane as N increases to infinity.