Bending and drilling aluminum plug plates
Digital Sky Survey Telescope Technical Note
19910912
Walter
A. Siegmund
Introduction
The optical design for the DSS telescope is a fast, simple optical
system with a nearly flat focal plane and low distortion. However,
focal plane curvature is such that an attempt must be made to
approximately place the ends of the optical fibers feeding the
spectrographs on the best focal surface rather than simply a plane.
Figure 1 is a graph of the best focal surface for spectroscopy
(Try54d, from the August 1991 NSF proposal). To further complicate
matters, the principle ray is only normal to the focal surface to
±30 mrad. Figure 2 shows the principle ray angle (with respect
to the normal to a plane at the focus) as a function of linear
distance from the center of the field. The situation is clarified in
Figure 3. The best focal surface and the paraxial surface, which is
normal to the principle ray, are shown together.
Figure 1. Best focal surface for the spectrographic mode. The
displacement of this surface from a reference plane is plotted as a
function of linear distance from the field center. The radius of the
3° field of view is 326 mm and the best focal surface deviates
by about 2.9 mm peak to valley from a plane. The coefficients of an
even polynomial fit to the raytrace results are given.
In the August 1991 NSF proposal, we describe one way to deal with
this problem. We deform the plate during drilling so that all the
holes can be drilled with their axes parallel. Each hole is drilled
with a shoulder at a precalculated depth. The fiber plugs are
inserted into the holes and registered against the shoulders. On the
telescope, the plate is allowed to relax flat, and if the plate was
deformed properly during drilling and the shoulders located
correctly, the tips of the fibers should lie along the best focal
surface and the fibers should be aligned with the principle ray.
Figure 2. Principle ray angle (with respect to the normal to a
plane) as a function of linear distance from the center of the field.
The coefficients of an odd polynomial fit to the raytrace results is
given.
In this approach, about 3 mm of plate thickness is allocated to
accommodate the range in depths of the locating shoulders (2.9 mm).
An additional 3 mm of plate thickness is needed for plug designs that
reference the fiber angle to the hole axis. (The exception is the
magnet plug approach which references fiber angle from the flat
shoulder surface. It requires that steel be used as a plate material
and that the shoulder bore be oversized to get good magnetic
retention.) The resulting plate thickness is 6 mm. However, 3 mm
thick aluminum plate is adequate to provide adequate stiffness
against gravity and forces exerted by the fibers. Currently, we
estimate that 3 mm aluminum blanks will cost $120k and that the
drilling of 3000 plates will cost $600k. The cost of blanks scales
linearly with thickness. Drilling may scale faster than linearly if
bit clogging in the deeper hole is a problem. These costs could
easily become dominant if thicker plates are used.
Other considerations include the fact that less storage space is
needed for 3 mm plates. Less massive bending drilling fixtures are
required. The bending strain is linear with material thickness and is
higher for thicker plates. Finally, the stress in the plate in the
telescope tends to reduce the deflection of the plate due to out of
plane forces.
In this note, I explore a different approach wherein the plate is
deformed both for drilling and use in the telescope. The original
plate surface is used for locating the plugs axially, i.e., no
shoulder is produced during the drilling process.
Figure 3. The best focal surface, the paraxial surface (the
surface normal to the principle ray) and the shape of an ideal
mandrel are shown. The mandrel is the shape that the plate should
have for drilling if all holes are drilled with their axes parallel
and if the plate is deformed to match the best focal surface in the
telescope. Thus, the mandrel curve is the difference between the
other two curves.
Model
An axis symmetric solid element model was used to investigate the
bending of the plate. This approach, in effect, reduces a
3-dimensional axis symmetric problem with axis symmetric loading to a
two dimensional problem. Solid elements are used. In two dimensions,
these are rectangular, but represent three-dimensional annuli with
rectangular cross-sections. The mesh is 89 elements wide by 5 high.
The material is aluminum 2024, 3.18 mm thick and 419 mm in radius. To
match the focal plane shape, displacement constraints were applied at
343 mm and 384 mm radius to force the plate to match the focal plate
slope at the edge of the field. A further displacement constraint was
applied at radii of 0, 50, or 100 mm to force the plate sag to match
that of the best focal surface. Without the latter constraint, the
plate sags about 0.5 mm too much. To match the shape of the ideal
drilling mandril, similar constraints are used. However, only radii
of 0 and 50 mm for the central constraint were examined.
Figure 4. Axis symmetric finite element model showing constraints
(triangles) used to force plate to match the telescope best focal
surface. The model consists of 5x89 mesh of axis symmetric solid
elements. The plate radius is 419 mm (16.5") and the thickness is
3.18 mm (0.125"). The constraints are applied at radii of 384 mm
(15.1") and 343 mm (13.5"). An additional constraint is applied at
the center of the plate.
Results
The best match to the best focal surface occurred with the central
constraint applied at a radius of 100 mm (Figure 5). For this case,
the best focal surface is matched to an accuracy of 50 microns peak
to valley. With the constraint applied at the center, the best focal
surface is matched to an accuracy of 250 microns peak to valley. By
way of comparison, with no constraint at the center, the mismatch was
830 microns peak to valley. Table 1 gives the RMS fit error of the
deformed plate to the best focal surface. Only model plg41-3 (central
constraint applied at a radius of 100 mm) satisfies the error budget
of 25 microns that we have established for this item.
Model Central constraint Mean surface Fit error
Name radius (mm) (microns) (RMS)
plg31 0 -65.0 83.0
plg41 50 -16.0 46.0
plg41-3 100 4.7 7.9
Table 1. Quality of fit to the best focal surface. The mean error
and root-mean-square (RMS) error are weighted by focal surface area.
The RMS is calculated with respect to the mean surface.
Figure 5. The fit of three models to the best focal surface is
plotted. The central constraint was applied at three different
locations; r = 0 mm, r = 50 mm, and r = 100 mm. The fit with the
constraint applied at a 100 mm radius is excellent.
These results are fairly insensitive to variations in the
constraints. Figure 6 shows results for three models. Plg32 is the
baseline case. In plg31, the displacement of the outer constraint was
increased with respect to the inner constraint 10% increasing the
slope at the edge of the focal plane by the same amount. In plg33,
the thickness of the plate was increased 10%. Assuming that the
telescope is focused on a probe at 70% of the radius, the maximum
focus error does not vary much. Changes in the plate thickness would
have a larger effect if the bending were done with forces rather than
displacements. However force errors would have less effect than
displacement errors on plate shape.
Figure 6. Fit error sensitivity to two parameters is plotted.
Plg32 is the baseline case. In plg31, the displacement of the outer
edge constraint is increased 10% with respect to the inner edge
constraint. In plg33, the plate thickness is increased 10%. If the
telescope is focused using a probe at r = 230 mm, the maximum focus
error for each case is not changed much.
It appears to be an advantage to drill a plate with a simple three
axis machine, if this is possible. Such a machine produces holes with
their axes parallel. So the question is, can the plate be deformed
for drilling so that when it is deformed to match the best focal
surface and mounted in the telescope, the hole axes are aligned with
the principle rays everywhere? It turns out that this is possible as
shown in Figure 3. The desired shape for drilling is the difference
between the paraxial surface and the best focal surface and is
labeled "ideal mandrel" in the figure. Actually, it doesn't matter
much if the plate in the telescope does not conform exactly to the
best focal surface. The model of plg31 matches the slope of the best
focal surface to 2 mrads or better.
Figures 7 and 8 shows results for two models. The constraints were
basically the same as shown in Figure 4. The pair of edge constraints
were chosen to match the shape of the ideal mandrel at the field
edge. The central constraint was set to produce the overall sag of
the ideal mandrel. In one case, the central constraint was applied at
the plate center. In the other case, it was applied at a radius of 50
mm. The principle ray angle error was 12 mrad RMS for the first case
and 6.7 mrad RMS for the second. We have established an error budget
of 10 mrad RMS for this item, so the first case exceeds the error
budget somewhat.
One problem with plg34 is that the maximum tensile stress is 389
MPa (56 kpsi) at the plate center. Since the yield strength of
aluminum alloy 2024 T3 is only 340 MPa (50 kpsi), this is a problem.
For plg44, the maximum stress is 107 MPa (16 kpsi), well below the
yield strength of the material. Thus it appears that some attempt to
distribute the central force will be necessary.
Figure 7. Two attempts to produce the optimal shape for plate
drilling are shown (indicated by the "ideal mandrel" curve). In
plg34, the central force is applied at the center of the plate. In
plg44, the force is applied at a radius of 50 mm.
Figure 8. The predicted errors in hole alignment with the
principle ray are plotted for the models of Figure 7. The curves are
labeled with the radius of the central constraint. For r = 0, the
hole misalignment is 12 mrad RMS. For r = 50 mm, the error is 6.7
mrad RMS.
Table 2 gives the forces required for each model. The forces are
given in newtons/radian. The tabular values should be multiplied by
2p to get the total force on each annulus. The forces are modest but
will have to be considered in the design of the bending fixtures.
Model Central Inner edge Outer edge
Name Constraint force force
plg31 34.69 -976.0 941.3
plg32 26.99 -810.0 783.0
plg33 40.42 -1032.0 991.4
plg41-3 52.41 -1021.0 968.2
plg34 -293.70 1193.0 -898.9
plg44 -357.30 1370.0 -1013.0
Table 2. Forces applied to model in N/rad. To get total force on
annulus multiply by 2p.
Conclusions
It is a pity that an adequate match to the best focus surface does
not occur with the central force applied to the center of the plate.
Applying a force on an annulus with a radius of 100 mm is probably
not something we want to do, even by use of 4 to 6 tension rods
attached to the plate at discrete locations. However, if it could be
done, the plate could be drilled from the sky-facing side since no
counterboring would be needed. In this configuration, the plate could
be supported by a mandrel with the ideal shape and this would give
excellent control of the hole axis angles.
The simple scheme with the support at the center does provide a
fair match to the best focus surface. If this were used with the
counterbore scheme of the August 1991 NSF proposal, only 250 microns
of plate thickness would be needed to generate the locating shoulder
and 3.2 mm thick plate could still be used.
The constraint at the center does comes close to meeting our error
budget for slope. It is likely that it could be improved or that the
error budget could be modified. I should point out that the error
budget is very tentative at this point since little analysis has been
completed.
If the plate were used flat in the telescope, no moment would be
required at the plate edge and the reaction at the plate center would
be smaller. The result is a better match to the paraxial surface.
Increasing the thickness of the plate should help as well.