Mirror cell

The calculation

Enter your mirror: the tool works out how it sags under its own weight on each of the usual cells, puts the supports where the sag is least, and draws the triangles and bars.

The sag of the surface, after refocusing: red is high, blue is low, green the mean level. The dots are the supports, the dashed circle the secondary’s shadow.

Plate solving

The mirror is cut into rings of triangles and treated as a plate, shear included (Toshimi Taki’s elements, Sky & Telescope, April 1996). As in David Lewis’s PLOP, the plate is solved once per mesh ring, and each cell is then a sum of these solutions: the optimiser can move the supports thousands of times in a few seconds. The results were checked against PLOP, digit for digit.

What is not calculated here: the tilted mirror and its edge support (see the 3D finite elements tab), the compression of the glass above the supports (PLOP’s Z88), and spread supports (pads), taken as points. For a thick mirror, over 20% of its diameter, the plate is an approximation.

3D finite elements

The same cell when the telescope points lower: the weight is shared between the cell (the sine of the altitude) and the lateral support (the cosine), pads on the edge at the height of the centre of mass, at the bottom of the map. The calculation is solid (3D finite elements): at the zenith it gives a slightly smaller error than the plate of the Plate solving tab.

The 3D finite elements tab treats the mirror as a solid, through its whole thickness, at every altitude and with its edge support. The results were checked against CalculiX.

A central hub, as on many Cassegrains: resting, the mirror sits on a shoulder round its hole (the forces spread round the hole’s edge, at the back, not holding it from turning); bonded, the whole bore is held, neither sliding nor turning. Tilted, the lateral support by the hub pushes the top of the hole up over a 60° arc, or, bonded, holds the whole bore (pushing at the top, pulling at the bottom). A mirror held only at its centre sags a lot at its edge: compare it with a cell.

A cellular blank, open at the back: hexagonal pockets (one at the centre) under a front plate of even thickness, inside an edge wall. Its plan is triangulated along every pocket’s outline, then extruded: the ribs from the back up to the pockets’ floor, the plate above. The supports sit under the ribs, where three meet: the sag is solved once for a support under each junction, and the cell is looked for among the junctions set symmetrically as its rings are, those with the least error. Depending on the cell, the blank sits one way or the other against its axes (turned by 30°): the best is kept.

What is not calculated: spread supports (pads), taken as points (hence local spikes on the 3D view, capped), slings, friction and play.