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-18(1 - v)DA33

(7.18f)

With Sphe 3 mode, the uniform loading is q = 64,DA40. The axial forces Fa,k and Fc,k to apply at r = a and at the end r = c of radial arms clamped onto the edge, are obtained by the static equilibrium relationships rn(2k-1)/km

Jn(2k-3)/km with k = 1, 2, ..., km for a MDM having km arms. The co-addition of various modes is obtained by summing the corresponding forces. The resulting forces to apply Fa,k and Fc,k are

n,m modes n,m modes

7.4 Degenerated Configurations and Astigmatism Mode 7.4.1 Special Geometry for the Astigmatism Mode

Special configurations for monomode flexures with m = n > 2 can be derived where forces Fa,k = 0, thus proving important practical simplifications in the deformable design by reducing the number of action points to a minimum (see the case m = n = 3 of triangle mode in Sect. 7.9). Because only requiring use of Fc,k forces, we call them degenerated configurations.

One restrains hereafter to the case of a flexure purely corresponding to third-order astigmatism z22 = A22r2cos29, i.e. Astm3, and determines the associated arm geometry of degenerated vase or meniscus forms (fifth-order triangle mode - or Tri5 mode - treated in Sect. 7.9). From (7.13a) and (7.13b), the condition Fa,k = 0 is satisfied if

Using the expressions of Mr (b, 0) and V(b, 0) given by (7.12d) for Astm 3 mode, the results from calculation show that such degenerated configurations can be obtained if the radial arms have the special geometry [38]

which is valid for flat to moderately curved vase or meniscus mirrors.

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