## W

where sinc x = sin nx/ nx. Compared to the classical parallel push-pull force distribution defined by equations (8.44), the above distribution acts with larger forces at the vertical mid-plane of the mirror whilst the forces in the regions y ~ ±a are much smaller.

A somewhat different alternative leads to redefining the width of the slices such that all the forces Fx,n are set equal. Such a variable y-spacing parallel push-pull force distribution was proposed by Mack (Fig.8.11-A) and adopted, with Nl = 24 parallel push-pull forces generated by direct astatic levers, for the plano-concave mirror of the 4.2 m Uk-Wht.

• Skew surface of forces: A second proposal by Mack [45] takes under consideration the fact that, for any plano-concave or meniscus mirror, the individual centers of gravity of the vertical slice elements are not at a same Z-value. Thus, constraining

Fig. 8.11 Variable y-spacing parallel and equal push-pull forces. (A) Top view of the plane surface of forces passing through the center of gravity of the mirror. (B) Top view of the skew surface of forces in which the individual centers of gravity lie [69]

Fig. 8.11 Variable y-spacing parallel and equal push-pull forces. (A) Top view of the plane surface of forces passing through the center of gravity of the mirror. (B) Top view of the skew surface of forces in which the individual centers of gravity lie [69]

the equal and parallel forces to lie in the skew surface which is defined from vertical lines passing through the individual centers of gravity (Fig. 8.11-B), it is shown from finite element analysis that, compared to the case where the forces act into a plane surface passing through the mirror center of gravity, the flexure can be reduced by about a third.

Use of a skew surface of forces allows us to introduce an interesting free parameter for the optimization of lateral mirror supports.

• Push-pull force distribution with increased shear: Schwesinger [73, 74] investigated force distributions where the tangential component is a free parameter. Denoting Fr,n and Frj the radial and tangential components of the applied lateral forces, where from Pythagoras F2n + F2n = F2n + Fyn, various distributions can be compared together by use of the fraction ¡5 of the weight W supported by the tangential forces Ft,n. Thus, ¡5 can be determined from

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