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Quasi-simply supp. edge

U 2

V(3+v)/(1+v)

4/(1+v)

Since Poisson's ratio v e [0,1/2], the zero power ellipse would be at 1.527 < U < 1.732 for a quasi-simply supported edge; with v = 1/4, this gives U = x/ax = 1.612 if y = 0. For reflective Schmidt systems, we see that the optical position of the zero power ellipse is intermediate between clamped and quasi-simply supported edge.

• Flexure of an elliptical plate in a closed biplate form: For obtaining k = 3/2 profiles, i.e. k = 1, let us consider a closed biplate form (Fig. 5.16) as the primary mirror of a reflective Schmidt system. This form is constructed with two identical plates of equal thickness linked together at their elliptical contour C with a cylinder. For continuity reasons, the ends of the elliptic cylinder are rounded along C in order to transmit the bending moment to the plates. Taking into account the effect of stress concentration due to a right angle junction, it can be shown that the exact geometry of a plate clamped into an infinite size corner is provided by a quarter-circle junction whose radius of curvature is equal to the thickness of the plate.

Fig. 5.16 Closed biplate form generating affine elliptical flexures by uniform loading only. The elliptical contour C of the plates is linked with a cylinder. The radial thickness of the cylinder realizes a semi-built-in condition and is varying along C to satisfy the best optical profile K = 1 (i.e. k = 3/2). In the (z, x) and (z, y) sections, the principal radial thicknesses of the outer cylinder are tx = bx — ax and ty = by — ay, respectively

In the aspherization process, starting from a plane mirror, the flexure directly provides the optical figure if

Rjooi = Zopt + pZEiass = 0 and k = 3/2 i.e. k = 1, (5.50)

where p= ±1 depends on considering an in situ stressing or a stress figuring. Assuming an in situ stressing, thus taking the positive value, the flexure is represented by [k = 1 in (5.49)]

1 x2 y2

Ux Uy

0 0