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Fig. 10.8 Thickness distributions T(x) of three cylinders generating a x2 flexure mode. Radial reactions at edges in equilibrium with uniform load, Fp = F—p = qL/2. Poisson's ratio v = 1/4. Middle thickness ratio t (0)/a = 1/10. Length parameters p = 0.45, 0.90, 1.35. Mirror aspect ratios L/a = 1.092 p

Fig. 10.8 Thickness distributions T(x) of three cylinders generating a x2 flexure mode. Radial reactions at edges in equilibrium with uniform load, Fp = F—p = qL/2. Poisson's ratio v = 1/4. Middle thickness ratio t (0)/a = 1/10. Length parameters p = 0.45, 0.90, 1.35. Mirror aspect ratios L/a = 1.092 p

An axisymmetric tubular shell with free edges is by itself in a static equilibrium when a uniform load q is applied all over its surface. Returning to (10.19) and using (10.14), we may introduce the expression of the shearing force Qx. Thus we obtain dM = - dX(DdW) = Q- (10.35)

and from (10.22), the general differential equation of the flexure of a cylinder may be expressed in a function of the shearing force,

dx a2

Using the dimensionless quantities T and W for the thickness and the flexure, equations (10.25) lead to

dx C

If q is the only external force applied to the cylinder, then the shearing force Qx is null,

so the first left-hand term in the differential equation vanishes. This result is a straightforward inverse proportional law,

^ If an axisymmetric cylindrical shell is uniformly loaded all over one of its surfaces, and if no other external forces are applied, then the thickness T and flexure W are reciprocal functions.

This law is of fundamental importance for the aspherization of X-ray mirrors because the boundary conditions at both edges vanish. Therefore only sliding contacts must be set at the end faces to prevent from pressure leak.

1 ^ Single-term parabolic flexure: Let us consider a parabolic flexure where the central section plane of the cylinder cannot extend or retract. This case corresponds to a dimensionless flexure represented by a single quadratic term,

W = C2 Ew = X2, X = x/a. From (10.38), the dimensionless thickness is simply

Hence the thin shell theory provides a valuable solution that is with infinite thickness at the plane of central section (Fig. 10.9).

2 ^ Two-term parabolic flexure: In the latter case of a single-term flexure, although the infinite thickness at the symmetry plane may be rendered finite by an optical tolerancing criterion, this difficulty can be easily circumvented by considering that a radial strain may appear all along the cylinder.

Hence if a parabolic flexure is generated from the origin to the edge x = ±ft, then in order to avoid singular poles at the cylinder edges - which, from the inverse proportional law, entails local infinite thicknesses -, it is necessary to

ÏV = X

/

\

c2

= 0.015 <7

0 0

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