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1 As in Sect. 1.13.10 and the other chapters, the positive sign convention for the bending moments Mr and Mt provides a more logical representation of the flexure than the negative sign convention currently used by other authors: as shown in Fig. 7.4, a positive curvature mode Z20 is generated by a positive bending moment Mr for a positive x-value. N.B.: For authors using the negative sign convention, an erroneous definition of Mrt is often encountered (see footnote in Sect. 3.2).

d3 Rnm + 1 d^Rnm _ 1 + vm2 dRnm + (1 + v)m2 R ' dr3 r dr2 r2 dr

The above set allows us to determine Bnm,Cnm,Dnm, and Enm as functions of Anm and then the bending moment Mr(b, 9) and net shearing force Vr(b, 9) distributions to apply at the ring edge r = b.

• First Clebsch-Seidel modes: For the first Clebsch-Seidel modes, the substitution of each znm mode into eqs. (7.11) and solving the associate system set leads to the following relationships.

B20 = (1 - y)(1 + v)(1 - ln a2)a2A2o/2 C20 = (1 - r)(1 + v)a2 A20 D20 = [2 - (1 - y)(1 + v)]A2o/2 E20 = 0

Mr (b, 0)= D2 [- ( 1 - v)C2o/b2 + 2(1 + v)D20 + (3 + v)E20 + ( 1 + v)E20 ln b2]

Spherical aberration 3rd-order mode - Sphe 3, n = 4, m = 0, with q = 64D1A40,

B40 = {v + r(5 - v) - [(1 + v) + y(1 - v)] lna2}a%0 C40 = 2 [(1 + v) + y(1 - v)]a4A40 D40 = [1 - v- y(5 - v + 4lna2)]a2A40 E40 = 8ya2A40

Mr (b, 0)= D2 [- ( 1 - v)C40/b2 + 2(1 + v)D40 + (3 + v)E40 + ( 1 + v)E40 ln b2] Qr (b, 0) = -4 D2E40/ b

B31 = ( 1 - y)[3 + v - (1 - v) ln a2]a2A31 /2 C31 = -(1 - y)(1 + v)a4 A31/2 D31 = yA31

E31 = (1 - y)(1 - v)a2A31 Mr(b, 0) = D2 [2(1 - v)C31/b3 + 2(3 + v)D31b +(1 + v)E31/ b] Qr(b, 0) = -2 D2[4D31 - E31/b2]

Vr(b, 0) = -D2[-2(1 - v)C31/b4 + 2(5 - v)D^1 - (1 + v)E31/b2] (7.12c)

B22 = [4 + (1 - Y)(1 - v)]A22/4 C22 = -(1 - Y)(1 - v)a4A22/12

Mr(b, 0) = 2D2 [(1 - v)B22 + 3(1 - v)C22/b4 + 6D22b2 - 2VE22/ b2] Qr (b, 0) = -8 D2 [3D22 b + E22/b3]

V(b, 0) = -4D2 [(1 - v)B22/b - 3(1 - v)C22/b5 + 3(3 - v)D22b

B42 = 3(1 - y)(3 - v)a2A42/4 C42 = -(1 - Y)(1 + v)a6A42/4 D42 = [Y- (1 - Y)(1 - v)]A42/4 E42 = -3(1 - y)(1 - v)a4A42/4 Mr(b, 0) = 2D2 [( 1 - v)B42 + 3(1 - v)C42/b4 + 6D42b2 - 2vE42/b2] Qr (b, 0) = -8 D2[3D42b + E42/b3]

Vr (b, 0) = -4 D2[(1 - v)B42/b - 3(1 - v)C42/b5 + 3(3 - v)D42b

B33 = [2 + (1 - y)(1 - v)]A33 /2 C33 = -(1 - Y)(1 - v)a6A33/8 D33 = -3 (1 - y)(1 - v)a-2 A33/8 E33 = 0

Mr(b,0) = 2D2[3(1 - v)B33b + 6(1 - v)C33/b5 + 2(5 - v)D33b3

+ (1 - 5v)E33/b3] Qr (b, 0) = -24 D2 [2D33 b2 + E33/b4]

Vr(b,0) = -6D2[3(1 - v)B33 - 6(1 - v)C33/b6 + 2(7 - 3v)D33b2

• Monomode forces Faand Fc,k : In order to generate the bending moments Mr and net shearing forces Vr at r = b, we may remark that the MDM design gains in compactness by applying axial forces at r = a and r = c instead of at r = b and r = c. With this choice, the axial forces denoted Fa,k and Fc,k are determined from the statics equilibrium equations (cf. Fig. 7.1)

Jn(2k-3)/km with k = 1, 2, ..., km for a MDM having km arms.

• Resultant multimode forces Fa,k and Fc,k : The forces Fa,k and Fc,k are determined for each mode by solving this system. The co-addition of various modes is obtained by summing the corresponding forces. The resultant forces Fa,k and Fc,k to apply to the MDM are

nm modes nm modes

With km = 12 arms, Table 7.1 displays the geometrical parameters of a metal MDM and the associated intensities of Fa,k and Fc,k forces for some Clebsch-Seidel modes.

A first 12-arm prototype MDM was built in a Fe87Cr13 stainless steel alloy (Fig. 7.2).

A diagram showing the distribution of Clebsch-Seidel modes of the optics triangle matrix and some He-Ne interferograms obtained with this MDM are displayed in Fig. 7.3.

Fig. 7.2 View of the 12-arm vase form and plane MDM. Geometrical parameters are a = 80mm, b/a = 1.25, c/a = 1.8125, ii = 4 mm, t2/ti = I/71/3 = 3. Elasticity constants of quenched stainless steel Fe87 Cr13 are E = 2.05 x 104 daN .mm 2 and v = 0.305. Deformation modes generated by rotation of differential screws at r = a and r = c. Air pressure or depressure can be applied onto rear side of clear aperture r < a for generating the Sphe 3 mode [Loom]

Table 7.1 Axial distribution of forces Fa<k and FCik of a 12-arm plane MDM (km = 12). Fe87 Crl3 stainless steel. E = 205 109 Pa. v 0.305. designed with t\ = 8 mm, 7= (i]/r2)3 = 1/27, a = 100mm. b/a = 1.24 and c/a = 1.6. The PtV amphtude of each Clebsch-Seidel models w = 0.1mm i.e. withA20 = w/a2, f"

A40 = w/a4, A31 = w/2a3. A22 = A20/2. A42 = A40/2. and A33 = A31 [Units: daN] ? ---

Angle Arm Cvl Sphe 3* Coma 3 Astm3 AstmS Tri 5 3

nb. n = 2 m = 0 n = 4 m = 0 n = 3 777 = 1 « 2 m = 2 n = 4 m = 2 n = 3 m = 3 2

0 * ^ ^ Fc^t i7«,^ FaJc Fc<k Fa<k Fcjc Fa,k FcJc I

0 0

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