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a while the net shearing forces Vx{±ax, 0} and Vy{0, ±ay} remain unchanged since the dMnt/ds component of the co-added mode in k of (5.43) is zero.

Considering a vase form, and denoting z1 the previous flexure for the inner plate, the outer ring of finite geometry could provide a rotation around its neutral axis. This would allow us to set the null powered zone at the location defined by the optics rule k = 3/2 in the axisymmetric case z ^ 2kp2 - p4 leading to 2k = 2 + k, which corresponds here to k = 1. We can represent this elliptic null power zone by

Assuming an elliptical clear aperture (a—, ay) linked to the inner side of the ring, could the ring generate independently a part of the quadratic deflexure? Its flexure z2 cannot be represented by functions of 1 - U2 for the two logarithm terms - that would correspond to r2 ln r2 and ln r2 of a circular ring - because the bilaplacian equation V4z2 = 0 would not be satisfied, i.e. V4ln U = 0. So, the flexure of an elliptical ring cannot provide affine curvilinear ellipses without requiring special moments and forces at the contour in addition to the uniform load reacting along it. However, an alternative consists of the use of two vase forms linked together with an outer elliptic cylinder in a closed shape (see hereafter). This avoids requiring Fa,k and FCkk forces as treated in Sect. 7.6.2.

The case of an elliptical plate with a simply supported edge can be envisaged from the case of a circular plate. It would correspond to k = 4/(1 + v) and z = z0 (1 - U2)(5+V - U2) , (5.46)

which would provide a null powered zone along the elliptical line defined by o 3 + v

The boundary conditions of a simply supported edge should correspond to a curvilinear bending moment Mn = 0 and a net shearing force Vn = 0 at the contour. This is not exactly satisfied since

Compared to the clamped edge case and with the incidence angle i defining the el-lipticity ratio by ay/a— = cos i, we see that these moments are small. Nevertheless, except for the clamped edge case, for a simply supported edge or quasi-simply supported edge obtaining a flexure whose contour in a plane strictly requires additional bending moments Mn and net shearing forces Vn applied to the outer cylinder by an appropriate external force distribution.

Various flexures of an elliptical plate of interest in optics can be represented by

Z = 8D 3a4 + 20212 + 3a4 <1 " 1 " ' K = 2(k — ^' (549)

where the associated parameters k and k determining the location of the null power zone are listed as follows.

Flexure geometry |
k |
Null power zone U |

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