## ZFMW zowM EM84c

where the first right-hand fraction is only material dependent. The reciprocal fraction allows defining merit-factors for equal mass stiffness as

• The four stiffness criteria: From (8.4b) and (8.4d), the four stiffness merit-ratios may be written in the form

each ratio depending on whether comparisons are with equal volume, equal mass, external or internal gravity bending forces. The higher these ratios, the less the flexure and the higher the resonance frequency of the fundamental mode. The stiffness merit factor for equal mass substrates, SF,M = E/v3/2, is an average value of the three latest merit factors. Furthermore, this mean stiffness is convenient for comparisons of equal mass mirrors to wind buffeting vibrations (Table 8.2).

8.4.2 Mirror Materials and Elastic Deformability Criterion

In the application of active optics methods, such as stress surfacing aspherization or in-situ deformation, important features are the choice of a material substrate that shows a linear stress-strain law and the largest linear range possible. The combination of those two features is the so-called elastic deformability ratio. Glass, vitro ceramics, and silicon carbide possess this linear characteristic up to rupture; however the linear range of all these brittle materials is quite moderate partly because their maximum tensile stress is time dependent (cf. Sect. 5.2.5 and Table 5.2).

In contrast several metal materials exhibit stress-strain linearity over extended ranges.

It is well known from unidirectional rupture tests that for any material the tensile ultimate stress is much lower than the compression maximum stress. Let otmax be the tensile maximum stress of a material as an acceptable limit that must not be exceeded to avoid rupture or plastic strain (see data in Table 1.10).

In all generality, a local curvature may be elastically generated by stress distributions orr, ott arising through the thickness of a plate. In the thin plate theory, these distributions are linear over the thickness of the plate and have null values at its middle surface. In Chap. 2 on curvature mode flexures, where orr = ott = O in the thin plate theory [cf. (2.10) or (2.17b)], we have shown that the ratio \o\/E governs the deformation sag of a plate. If orr = ott, so the local principal curvatures differ, the maximum stress of them will give a limitation in the flexure when the loading intensity is increased. Hence the corresponding maximum acceptable ratio for obtaining a maximum flexure - or elastic deformability ratio of a material (cf. (2.48d)) - is

where the significance of suffix F, V is that the bending is generated by a load F which is external to the plate and that the sag is for a given volume V determined by the diameter d and thickness t; these two latter dimensions must be the same for comparing materials elastic deformability in all generality (Table 8.2). The Mohs' hardness - as classically defined in the range 1 to 10 - of some linear stress-strain materials is included in the last column of Table 8.2.

### 8.5 Axial Flexure of Large Mirrors Under Gravity

Important features for large telescope mirrors are the concept and design of its axial support system. In general the quasi-perfect concept of a mirror support by air pressure is avoided for technical reasons, so the remaining support system options all use a discrete pad distribution. An optimal design of the axial support system consists in the determination of the minimal number of pads for which the flexure under gravity will satisfy a convenient optics tolerance such as, for instance, a diffraction limited criterion.

### 8.5.1 Density Distribution of Mirror Support Pads

A convenient investigation for the determination of the flexure of a large mirror under gravity is to consider an infinitely large mirror where the supporting pads are equally spaced in a 3-fold symmetry of step 2a. As a preliminary parameter, one requires defining the pad density at the back surface of the mirror (Fig. 8.4).

The parallelogram in the figure contains four pads, thus the pad density per unit surface area is

From this definition, if the mirror is of finite diameter d, then the total number of axial support pads is n 2 n d2 d2

Returning to the case of an infinitely large mirror, one shall calculate hereafter the flexure of a mirror sub-element that surrounds a pad.

8.5.2 Flexure of a Mirror Sub-Element Supported by a Ring Pad

In the above 2-D distribution, the infinitely large mirror may be seen as an assembly of successive hexagon elements, each of them supported by a ring pad. To a sufficient approximation for determining the flexure of this hexagon element, one may assume that its contour is a circle (C) which is tangent to six proximate circles (Figs. 8.4 and 8.5).

inmm111 v t ,■ K | |

Ring pad ' w |
Wi | Contour circle (C) ->| |

Fig. 8.5 Circular mirror sub-element of an infinitely large mirror axially supported in a 3-fold symmetry by ring pads of radius b. The outer radius a of the mirror sub-element is the radius of circle (C) in Fig. 8.4

Fig. 8.5 Circular mirror sub-element of an infinitely large mirror axially supported in a 3-fold symmetry by ring pads of radius b. The outer radius a of the mirror sub-element is the radius of circle (C) in Fig. 8.4

Hence the problem of the flexure of a large mirror under gravity reduces to that of a circular mirror element, of outer radius a and thickness t, supported by a ring pad of radius b where the gravitational volume load is equivalent to a surface load q = Igt, (g < 0). If the thickness of the mirror is constant and if its curvature relatively large, then the flexure z(r) of the mirror "sub-element" is derived from Poisson's bilaplacian equation (see Sect. 1.13.10)

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