Moderate Axisymmetric Deformations of Optical Membrane Surfaces

Large deformable membranes are currently used as optical reflector surfaces for solar concentrators. In this paper, we study moderate axisymmetric deformations of optical membrane surfaces and their impact on the macroscopic optical quality of the deformed surface. A uniform pressure loading applied to an initially flat membrane surface is assumed in attaining the final concentrator surface shape. The problem is important in the design of focusing stretched-membrane heliostat modules. Insights into the design of stretched-membrane parabolic dish reflectors also result from this work. This paper presents several simple design and sizing relationships, based on a nonlinear variational principle, in terms of common engineering parameters. Good approximate predictions of the nonlinear load-deformation response and the stress state in the membrane are derived; simple but accurate predictions for membrane surface slope and nominal focal length also result. The phenomena that lead to deviation of the surface shape from the desired parabolic reflector shape are identified and design procedures to minimize those deviations are deduced. Results show that the macroscopic optical surface quality, for a given nominal f/D (concentrator focal length/diameter), is highly dependent on the non-dimensionalized membrane stiffness parameter, Et/T_o (membrane modulus × thickness/initial tension). Further, for the heliostat cases of interest, the focal-length variations relative to the desired parabolic shape were shown to be about ±5 percent about the nominal value.