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where ß = L/2a is the dimensionless semi-length of the cylinder. The curvature 1 /RCir of the figuring circle segment is, from (10.51) and z/a|Cir = z/a|Opt + z/a |Flex,

Now introducing an additional constant a2 to avoid the pole singularity, from (10.55), the dimensionless flexure becomes

W - cEW = ¿2 (a2 + 2ß2X2-X4) > 0, X € [-ß,ß]. (10.57)

From the inverse proportional law (10.38), TW = constant, the dimensionless thickness is t = 3,2 8CbR2 ^ (io-58)

a3(a2 + 2 ß2x2 - X4) which, from (10.50b), may be written

Optimizing the value of the constant C, an appropriate setting of the free parameters T(0) and a2 provides a thickness geometry of the mirror which then can be readily fabricated either in a linear metal alloy - satisfying Hooke's law -, or in glass, or in vitro ceram (Fig. 10.12-Right).

5 Minimal volume with an even biquadratic curve: In an axial section of a cylinder, let a curve g = p + qx2 - X4 be tangent to a straight generatrix at x = ± \/q/2. Setting the curve ordinate such that g(\Jq/2) = 0 entails p = -q2/4, so the curve is g = -1 q2 + qx2 -X4. In a 2n rotation about the cylinder axis, the volume generated by the curve and its tangent generatrix - the x-axis -, from the origin to the edge, is p v ~Jo (-1 q2 + qX2 -X4) dX = -4 q2P + 3 qP3 - 5 P'-

This volume is minimal when dv/dq = 0, which entails q = |P2. Hence the curve providing a minimal volume is g = -1P4+3 P2x2 - x4.

From dg/dx = 0, the intermediate extremum of this curve is at the abscissa x/P = 1/\/3 = 0.5773...

10.3 Elasticity Theory of Weakly Conical Tubular Shells

10.3.1 Flexure Condition for Pure Extension of Axisymmetric Shells

In the general case of an axisymmetric two-mirror telescope, let z(x) be a section representation of an optical surface where x = x/a0 is an axial variable with respect to the radius a0 of the mid-thickness of a shell at the origin x = 0. The function z(x) may either be polynomial or parametric as in this latter case for a WS telescope that strictly satisfies Abbe's sine condition (cf. Sect. 10.1.3). For instance, let us assume in the elastic aspherization that the figuring is conical while the mirror is under stress. Then, the radial quantity of material f (x) to be removed is opposite to the optical sag i.e. in the form f (x) = —z(x) + cix + c0. It is natural to set the constants c0, c1 such that f (j) = f (—j) = 0 at the mirror ends, thus giving the mirror sag with respect to the cone lying to the ends. However, these two conditions lead to radially unmovable ends and we have seen from the cylindrical shell theory that this entails use of radial reactions opposite to the load or infinitely thick ends. Therefore, it is preferable to consider hereafter the other alternative without any radial reaction to the load q.

The inverse proportional law, TW = constant (10.38), strictly applies to cylindrical or quasi-cylindrical shells. For a truncated weakly conical shell, with a mean slope angle up to a few degrees, another law will be derived in Sect. 10.3.3. Similarly as for a cylindrical shell, an appropriate constant a2 must be added to the quantity f (x) to be removed by stress figuring, in sort that no point of the mid-surface of a conical shell is with a null - or a too small - radial displacement otherwise this would entail a singular pole and thus an infinite - or too large - thickness.

In the sign convention, the flexure function W is positive when towards the conical shell axis; this corresponds to a retraction of the shell and a positive uniform load q. Hence we have the two following cases of monotonic sign flexure

Ew t

Ew t

where after stress figuring and elastic relaxation, the constant radial retraction - or extension - of the shell, due to the constant a2 term, vanishes.

If the function f(x) is a polynomial expansion, then it includes odd and even terms. We of course assume hereafter that the main conical term, determined by the slope of the mirror ends, is not obtained by stress figuring. Hence the flexure may be written in the polynomial form

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