would lead to

However, the radial and axial thicknesses of the cylinder should be infinite which then is rather an academic case.

6.5 Determination of a Variable Thickness Vase Shell 6.5.1 Flexure Representation in the Shell z,r Main Frame

In the previous sections the local orthogonal displacements wn, un at any ring element links r = rn are curvilinear quantities. Hence, the axial and radial displacements in the main coordinate frame (z, r) of the shell are

Szn = Wn costan-^<R>) + un sintan-1 (<R>), (6.55a)

Srn = — wn sintan-1 (<R>) + un cos tan-1 (<R>). (6.55b)

In this frame, the flexure zFlex of the middle surface is the locus of points of coordinates Szn, rn + Srn resulting from the displacements after deformation. Given an n-value, the corresponding flexure at this point is zFlex{rn + Srn} = Szn . (6.56)

Since the Kelvin functions are polynomial expansions for small or moderate argument values r/t, a polynomial smoothing can be obtained for the axial flexure in the form of the following even series zFlex = £ a2lr21, (6.57)

where, for a shell made of N linked elements, the determination of the a2i coefficients requires taking into account the flexure of each junction point, therefore solving an N-unknown system.

The shallow shell theory assumes that the largest of thicknesses {tn } is negligible compared to the mean radius of curvature <R > of the shell. In accordance with this condition, we will consider in the next sections | tn/<R > | ratios equal to or lower than 1/100. Furthermore, we assume that the curvilinear thicknesses {tn} and the associated thickness tz(r) measured along the z-axis remain unchanged during stressing so the flexure of the shell outer surfaces are the same as that of the middle surface.

6.5.2 Inverse Problem and Thickness Distribution

Given a vase shell with a discrete thickness distribution {tn} made of N meniscus rings and an outer cylinder N + 1, a square 4N-matrix similar to (6.36) allows us to solve the Ci,n, Mr,N+1, and Nr,N+1 unknowns. Then the associated displacements {wn,un} are known which determines the flexure zFlex in the form (6.57). To summarize, the successive calculations are

Now given a flexure zFlex(r) to be achieved in an even expansion form the inverse problem is to determine the associated thickness distribution {tn}. In general, starting from a constant thickness shell, such a distribution can be determined by an iterative code.

A dedicated code for this purpose was elaborated on the basis of variational parameters where the coefficient ratios a6/a4, a8/a4, a10/a4, and a12/a4 in flexure (6.57) form the target. After preliminary calculations for determining a correction vector, iterations toward the required coefficient ratios are carried out by linear algorithms which modify the {tn}-set for n > 1 up to obtaining the convergence. The iterations also provide a correction factor for the load; then the whole process is repeated with a new t1-value up to obtaining the required load q. In a final stage, the curvilinear solution {tn} is transformed into a tz(r) axial function for practical realization of the thickness distribution on the shell meniscus rear side.

6.6 Active Optics Aspherization of Telescope Mirrors 6.6.1 Active Optics Co-addition Law

We consider here the case of an optical surface which can be aspherized by active optics as a result from the co-addition of a flexure and a spherical surface. The position of the problem is to determine the curvature 1 /RSphe of this sphere and also the associated parameters for the flexure. These parameters are the curvature 1 /RFlex, the mirror thickness distribution t(r), the uniform load q, and the boundary conditions. This concerns the above inverse problem.

The aspherization process that generates the optical figure zopt by flexure zFlex of a spherical surface zSphe results from the active optics co-addition law zopt = zSphe + zFlex , (6.59)

from where a basic theorem can be stated as follows.

^ Whatever the aspherization process, either

• in situ stressing after spherical figuring without stress, or

• elastic relaxation after spherical figuring while stressed, the sphere and also the flexure are algebraically the same for obtaining the same optical figure. The uniform loads q have opposite signs and equal absolute value.

In the second process only the sign of the flexure during stressing is opposite to the above zFlex. However, the sign of the flexure zFlex is the same for in situ stressing or elastic relaxation.

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