Surface Accuracy Analysis of an Inflatable Rectangular Membrane of a Space Reflector

Abstract

The surface accuracy of an inflatable rectangular membrane subjected to pressure and isotropic tension in a reflector is investigated. The inflatable membrane is assumed to be placed at an arbitrary location in a reflector with an arbitrary rotation angle α between the side of the membrane and the principal direction on the ideal parabolic surface. The edges of the membrane are assumed to coincide with an elliptic or a hyperbolic paraboloidal surface. The deflection of the membrane is obtained analytically based on the linear equation (Poisson's Equation) for a pressurized membrane. Different kinds of surface error optimizations with respect to the pressure/tension ratio q and parameters that determine the edge deflection of the membrane are carried out. The effects of q, α , the aspect ratio λ and the location of the membrane upon each surface error of the optimum membranes are examined. The surface error of an inflatable rectangular membrane whose edges coincide with the approximate parabolic surface (APS) of a reflector is found to be zero on the assumption that the appropriate pressure/tension ratio is applied. With the exception of an inflatable flat-edge-optimum membrane, each surface error of inflatable optimum membranes takes a value of zero or increases monotonically as the distance from the vertex increases, and the surface error also takes a value of zero or constant, or decreases as λ increases or decreases away from one. Moreover the surface error is found to be well below that of the unpressurized optimum membranes when the membranes are located near the vertex of the reflector.

Effects of Tension and Edge Deflection upon the Surface Accuracy of a Rectangular Membrane of a Space Reflector

“…By leaving terms of 0 to 2nd order with respect to ξ and η , and considering the coefficients excepting a 20 and a 02 are zero, we can approximate the ideal parabolic surface by APS ζ , which is expressed by the following equation:…”