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This paper presents the results of numerous design studies carried out at Perkin-Elmer in support of the design of large diameter controllable mirrors for use in laser beam control, surveillance, and astronomy programs. The results include relationships between actuator location and spacing and the associated degree of correctability attainable for a variety of faceplate configurations subjected to typical disturbance environments. Normalizations and design curves obtained from closed-form equations based on thin shallow shell theory and computer based finite-element analyses are presented for use in preliminary design estimates of actuator count, faceplate structural properties, system performance prediction and weight assessments. The results of the analyses were obtained from a very wide range of mirror configurations, including both continuous and segmented mirror geometries. Typically, the designs consisted of a thin facesheet controlled by point force actuators which in turn were mounted on a structurally efficient base panel, or "reaction structure". The faceplate materials considered were fused silica, ULE fused silica, Zerodur, aluminum and beryllium. Thin solid faceplates as well as rib-reinforced cross-sections were treated, with a wide variation in thickness and/or rib patterns. The magnitude and spatial frequency distribution of the residual or uncorrected errors were related to the input error functions for mirrors of many different diameters and focal ratios. The error functions include simple sphere-to-sphere corrections, "parabolization" of spheres, and higher spatial frequency input error maps ranging from 0.5 to 7.5 cycles per diameter. The parameter which dominates all of the results obtained to date, is a structural descriptor of thin shell behavior called the characteristic length. This parameter is a function of the shell's radius of curvature, thickness, and Poisson's ratio of the material used. The value of this constant, in itself, describes the extent to which the deflection under a point force is localized by the shell's curvature. The deflection shape is typically a near-gaussian "bump" with a zero-crossing at a local radius of approximately 3.5 characteristic lengths. The amplitude is a function of the shells elastic modulus, radius, and thickness, and is linearly proportional to the applied force. This basic shell behavior is well-treated in an excellent set of papers by Eric Reissner entitled "Stresses and Small Displacements of Shallow Spherical Shells".1'2 Building on the insight offered by these papers, we developed our design tools around two derived parameters, the ratio of the mirror's diameter to its characteristic length (D/l), and the ratio of the actuator spacing to the characteristic length (b/l). The D/1 ratio determines the "finiteness" of the shell, or its dependence on edge boundary conditions. For D/1 values greater than 10, the influence of edges is almost totally absent on interior behavior. The b/1 ratio, the basis of all our normalizations is the most universal term in the description of correctability or ratio of residual/input errors. The data presented in the paper, shows that the rms residual error divided by the peak amplitude of the input error function is related to the actuator spacing to characteristic length ratio by the following expression RMS Residual Error b 3.5 k (I) (1) Initial Error Ampl. The value of k ranges from approximately 0.001 for low spatial frequency initial errors up to 0.05 for higher error frequencies (e.g. 5 cycles/diameter). The studies also yielded insight to the forces required to produce typical corrections at both the center and edges of the mirror panels. Additionally, the data lends itself to rapid evaluation of the effects of trading faceplate weight for increased actuator count,
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