sabato, novembre 03, 2007

Misaligned sling suspension points

This is an analysis of the problem described by Nils Olof Carlin where a sling perfectly centered at the centre of gravity of the mirror is suspended to points that do not stand in the same plane. This causes the departure of the sling from the disk with a small angle that in turn produces a slight pull out of the mirror plane. The case may happen for improper manufacturing, but also as a consequence of mirror displacements during collimation.
The analysis here shown uses different assumptions than that of Nils: in this case the mirror is vertical and the departure angle is supposed to be 1/100 rad, which means a pull of 2.21 N (the load case is linear and one may proportionately evaluate the effect of smaller angles).
As shown the PTV surface error is 164 nm (1/1.67 waves on wavefront!!) mostly astigmatism (tilt and defocus are removed). This figure is somewhat exaggerated because the mirror is restrained by a simple three-point support on the back (for the marginal loads of the out-of-plane pull). A real 9 or 18 points support would produce reactions to the out-of-plane pulls somewhat better distributed and thus an overall lesser deformation. Nevertheless the conclusion is clear: not only the sling needs to be accurately placed under the centre of gravity, but also the suspension points need to be accurately on the sling plane.
If one supposes to be able to keep the departure angle within 1/1000 rad (the suspension points must lie within 0.38 mm from the plane assuming a base distance of 380 mm (the distance between point of departure and point of suspension), then the deformations would be 16.4 nm (1/17 waves).