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For a given mode znm, these results allow the determination of the rigidity and of the associated external forces to apply to the mirror surface and also the forces and moments to its contour. Hereafter and for reasons of practical application, we will mainly consider the two cases q = O and a uniform load q = constant.

As previously found for variable curvature mirrors i.e. the Cv 1 mode, the choice of the external load q = O or q = constant in (3.14) allows one to recover the single solution with D{a} = constant and Mr{a} = constant that belongs to the CTD class, as well as the three solutions with D{a} = 0 and Mr{a} = 0 that belong to the VTD class and characterized by the two forms called cycloid-like and tulip-like. The four rigidities described in Chap. 2, appear from the coefficients A0 (logarithm term), A1, A2, and the roots a1 = 0, a2 = -2.

The net shearing force Vr represents the resultant acting in z direction into the plate at a radius r. This force was first derived by Kirchhoff [11, 12, 27] when a twisting moment Mrt exists into the plate i.e. for the case of non-axisymmetric deformations. This force is defined by1

r 39

After substitution, the net shearing force is

Vr = - [(n - 2)n2 + m2 - v(n - 1)m2} A0 rn-3ln r cos m9

j (-\(n - 2)n2 + m2 - v(n - 1)m2l \ + £ V jA rn-3-a cos m9. (3.16)

• Boundaries and continuity conditions: In this chapter on the thin plate theory we assume that the "in-plane" forces at the middle-surface are negligible so the plate is free to radially move at the boundaries. Hence the four cases describing an edge boundary condition, or the link continuity condition between adjacent zones, at a contour of radius r = a are the following.

1. Built-in or clamped edge boundary f^) = 0. (3.17a)

2. Free edge boundary

1 The present definition of the net shearing force Vr is consistent with the positive sign convention of the three flexural moments in the above Eq. (3.1).

There is an error in Theory of Plates and Shells by Timoshenko and Woinowsky-Krieger at Eq. (j) p. 284: Their convention uses a negative sign convention in the definition of the three moments Mr, Mt and Mrt while the sign of their shearing forces Qr and Qt with respect to the Lapla-cian term is as above Eqs. (8a) and (8b), so that the two equations in (8c) are the same. Hence the correctly associated representation of the net shearing force should be Vr = Qr + d Mrt / (r d r), with their notation.

Several other authors as well use twk's negative sign convention or the present positive sign convention in defining the two bending moments, but the torsion moment Mrt appears with an opposite sign whatever convention is used. In order to respect the equilibrium equations of statics, the sign before Mrt is also changed in those equations so the Poisson biharmonic equation is satisfied. However, there is an error in the sign before dMrt/(rdr).

3. Simply supported edge boundary

4. Linked edges of adjacent zones 1 and 2 - continuity

Mr{a,e} |1 = Mr{a, 6} |2 and V{a, 6} |1 = Vr{a,0} |2. (3.17d)

In the next sections are presented the rigidities and associated force configurations providing the third-order aberration modes. First, for a given mode n, m, the rigidity is derived from the choice of the loading q in (3.14) which provides solutions for the coefficients ai and the existence or not of the A0 term. Then, the global form of the rigidity is known from (3.9) by combining all A0 and Ai terms. The bending moment Mr and net shearing force Vr, given by (3.11a) and (3.16), define the possible associated configurations of external forces and boundaries allowing one to select compatible arrangements of A0 and Ai terms, therefore, providing the final rigidity for each distribution.

3.3 Active Optics and Third-Order Spherical Aberration

The third-order spherical aberration, Sphe 3 mode, is defined by n = 4 and m = 0. This axisymmetric wavefront function is represented by

The solutions ai to consider must be of the simplest form as possible, that is, for instance, a configuration of two loads in equilibrium, defined by a combination of two of the three following elements: an axial force at center, a uniform load onto the mirror surface, a constant force along the contour.

Considering (3.14), the two first terms in ln{r} and in r0 cannot provide simultaneously q = 0 or q = constant except if AO = A0 = 0 and q = 0, so that no component in A0lnr is possible for the rigidity. From (3.11a), (3.16), and (3.14), the remaining part of Mr, Vr, and q are respectively

Vr = Qr = 4A40 j [-8 + (3 + v) a,] A, r1-a , (3.19b)

j q = 4A40 j [(3 + v) af - 2 (7 + v) * + 16] Ai r-ai. (3.19c) i=1

The roots are j for q = 0 ^ a1 = 8/(3 + v) and a2 = 2,

Therefore, substituting these ai in (3.9), all the distributions of flexural rigidity are contained in the representation

Let us write the rigidity as

with

where the rigidity D0 is a constant used for normalization, and C1, C2, C3 are di-mensionless coefficients. SinceD = Et3/12( 1 — v2), a dimensionless thickness T40 can be similarly defined from a constant t0, the scaling thickness, by

Substituting the roots a, in (3.19a) and (3.19b), the bending moment and net shearing force are

Mr = 4(3 + v)A40 [Ax r2-8/(3+v) + A2 + A3 r2 Vr = -8A40

Depending on the possible arrangements, selection and value of coefficients A;, various configurations allow generating the Sphe 3 mode in the CTD and VTD classes.

3.3.1 Configurations in the CTD Class (A\ = A2 = 0J

Solutions in the CTD class are obtained if the coefficients Ai = 0 and A2 = 0 : these are plates or moderately curved meniscus when deformed by a uniform load applied to their surface. With a3 = 0, (3.19c) provides

0 0

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