W C4Ea P4 X4 X2 P21045

where P is the reduced length parameter such that x = ±P at the cylinder edges and, from (10.26), C4 = C is the unknown constant. The radial reacting forces Fp and F_p at the edges ensure that the radial displacement is zero at x = ±P. After substitution into (10.26), the dimensionless thickness T = t/a is a solution of

which, using similarly the variable u = T3, becomes

This equation may be solved by numerical integration from the thickness T(0) = t0/a = u01/3 at the origin. Successive iterations with X varying from zero to P, up to obtaining u(P) = 0, provide the unknown C4. The results would show that, compared to the parabolic flexure case, the thickness of the cylinder in the edge region is relatively smaller for the same flexural sag.

Similarly as the parabolic flexure case with radial forces Fp and F-p at the cylinder ends, and because of those forces, this configuration is a difficult practice for glass or vitro-ceram mirrors.

• Uniform load and free ends - Inverse proportional law: The inverse proportional law (10.38) states that if a uniform load q is applied to the cylinder surface and if no other external forces exist, then the thickness and flexure are reciprocal functions, TW = C = constant.

1. Single-term fourth-degree flexure: Investigating a fourth-degree flexure, where the central section plane of the cylinder cannot extend or retract, leads to the single-term representation W = x4. From (10.38), the dimensionless thickness is T = C4/x4 with x2 < P2. Similarly as in a parabolic flexure, the thin shell theory provides a valuable solution in the form of an infinite thickness at the plane of central section. However, this solution is mostly academic.

2. Two-term fourth-degree flexure: Assuming that the requested optical shape is of the form p4 - x4, the singular poles for the edge thickness, i.e. at x = x/a = ±P, are avoided if the flexure of the cylinder is set as

qa where a2 is a constant. From (10.38), the dimensionless thickness is, similarly to (10.42),

If the free constants a2 and C4 are appropriately chosen, the thickness geometry can be readily fabricated for glass or vitroceram mirrors (Fig. 10.11).

T(0) = 1/20

C4 = 0.225 a2+ßA = 4,5

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