assumed as that of M1 at r3,max. Therefore, the optical surface results from the sum of the following three terms z3,Opt = z3,Sphe + z3,Flex + z3,Rota • (6.78)

We know that a bending moment applied at the perimeter of a constant thickness plate (cf. Chap. 2) generates a purely quadratic flexure. Since we will see that the variation of the {tn} distribution for M3 is less than 1%, the flexural rotation caused by its link to M1 can be accurately taken into account by the single term representation

2r3,max V dr J r3,max

With N = 10 successive shell rings all included inside the continuous link at rN = r3,max, the analytical determination of the normal thickness distribution {tn} of the M3 meniscus made of rN/N radial width elements leads to a slightly increasing distribution from center to edge (Table 6.11).

From (6.78) and associated to this {tn} distribution, the co-addition law of the surfaces writes, in [mm], z3,sphe = 0-223988 10-3 r2 + 0-11238 10-10 r4 + 0-0113 10-16 r6 + 0^141 10-24 r8 z3,Flex = 0-001396 10-3 r2 - 0-86954 10-10 r4 + 0-1784 10-16 r6 - 0-141 10-24 r8 z3,Rota = 0-002178 10-3 r2

z3,Sum = 0-227562 10-3 r2 - 0-75716 10-10 r4 + 0-1897 10-16 r6 + 0-106 1 0-48 r8

z3,Opt = 0-227562 10-3 r2 - 0-75716 10-10 r4 + 0-1897 10-16 r6 + 0-000 1 0+00 r8

where the total radius of curvature of the flexure RFlex is deduced from the A2 terms of the sum zFlex + zRota of a perfectly built-in edge plus the tangential edge rotation of the intermediate ring assumed equal to that of the M1 mirror at rN = r3,max. The final design of the double vase shell is derived from each thickness distribution in the two latter tables (Table 6.12 and Fig. 6.20-Left).

The spherical figuring without stress was carried out at the same radius of curvature for both mirrors by means of a single full size tool. The finite element analysis of the shell when stressed at q = -0-80 x 105Pa gives maximal stresses

Table 6.11 Normal thickness distribution {tn} for the hyperbolization of MINITRUST vase shell M3 mirror f/6.1. Mirror clear aperture 180 mm. Strictly built-in at the contour. Load q=-80kPa. rN = 90mm. ROpt = 2,197.2mm, RSphe = 2,232.2mm, RFlex = 139,900mm, <R >=2,267 mm. [Units: mm]

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