Appendix A: Derivation of the Secondary Sag as a
Function of Telescope Altitude
One-half of the secondary support structure consists
of the two rods shown in Figure 9 plus the opposing rods on the other
side of the secondary cage. The total tension force on a single rod
can be decomposed into a thermal component and a base tension that is
set at the time when the rods are originally tensioned with the
turnbuckles. The base tension of a single rod is not a fixed
quantity, but is rather a variable which responds to the force
balance between all of the rods attached to the secondary cage. I
assume here that the cage is aligned symmetrically so that only half
of the rods need to be considered at a time in this force
balance.
The force on a single rod is given by
where
FT
= the base tension on the rod,
=
the thermal tension given by changes in the rod temperature from the
original rod installation, and
= the
tension changes in the rod owing to a variable gravity load as the
telescope changes orientation.
Note that the thermal changes are defined such that
positive values of
correspond to increases of rod temperature.
I will define "axial" forces as those in the direction of the
telescope primary. Radial forces are those perpendicular to the
telescope boresight. The axial force,
Fa, on a single
rod is given by
where a = the angle between the rod and the radial
direction. Figure 9 shows that a=13.6º for the bottom rods and
a=18.2º for the top rods. I will call these angles
a1 and
a2,
respectively.
To keep track of each rod I will use the subscripts
lb., lt, rb, and rt which refer to left-bottom, left-top,
right-bottom, and right-top, respectively. For example,
Flb denotes the
total force on the left-bottom rod and
FlbT denotes the
base tension on this same rod.
For convenience, I will also assume that the rods
considered are aligned with the horizon when the telescope is at zero
degree altitude. The end results derived here are independent of this
assumption. The alignment of the rods with respect to the gravity
vector only affects how the radial loads are distributed on the rods
at the horizon and the actual angular dependence of the sag with
altitude. It does not affect the total magnitude of the sag.
At the horizon all of the
forces will assumed to be zero. This can be
done because the starting focus point is arbitrary for this
calculation. The axial force balance equation is then given by
(1) 
At the zenith the gravity forces act in opposition to
the axial tension in the bottom rods and in parallel to the axial
force in the top rods. I account for this by writing the force for
the top rods as
and by writing the force for the bottom rods
as
With this convention the force of gravity is always
considered positive and the axial force balance at zenith
becomes
(2) 
where W is the weight of the secondary cage including
the mirror and electronics. I have assumed in the equation above that
the cage is support symmetrically by both halves of the support
system of rods. Combining equations (1) and (2) above gives
(3) 
Note that all of the thermal terms have canceled.
In other words, as long as there are no temperature changes between
the time the telescope is at the zenith and when it is at the
horizon, thermal changes will not alter the differential sag that
takes place as the telescope moves in altitude. This is true even if
all of the rods are not at the same temperature.
This, however, is not the whole picture
because this calculation does not tell you where the starting point
of the secondary lies, it will only give you the differential
position between horizon and zenith orientations. But, it is
instructive to finish the calculation that we have started before
moving on to address this issue. I will address the issue of the
absolute position of the secondary in
Appendix
B.
If the cage is stiff, then we can assume that all of
the rods feel the same gravitational force. In other words, we
have
We make this simplification in equation (3) and solve
for ђFg to
give
(4) 
The definition of stress gives us the equation
(5) 
where F is the force on a rod,
Arod is the
cross-sectional area of a rod, E is the elastic modulus of the rod
steel, 
is the length
change of the rod, and L is the original length of the rod. We can
substitute for F in
equation (5) if we interpret to be the change in length from the original tension
balanced position. Doing this, combining equations (4) and (5), and
solving for gives
(6) 
Equation (6) gives the change in length of a rod due
to gravity as the telescope moves from the horizon to zenith. We must
now convert this to the equivalent change in the axial position of
the secondary cage. Figure A1 shows the geometry
required to make this conversion and the derivation of the
equation
(7) 
which gives the relationship between the axial motion
of the secondary cage, 
, as
a function of the change in length of a support rod, . We now combine equations (6) and
(7) to solve for the total secondary sag expected as the telescope
moves from the horizon to the zenith. This gives
(8) 
were L is the length of any rod of your choice and a
is the angle of this rod to the direction of the telescope radius.
Note that if the cage were symmetric with all of the rod lengths
equal and a1 =
a2, then we would
have
(9) 
For the case of the 3.5-m, equation (9) can be used
in place of equation (8) with only minor errors (10%) if L is the
average length of both top and bottom rods and a is the average angle
of both top and bottom bars.
As a recap, the variables that go into equations (8)
and (9) are
L = the length of a secondary support
rod
W = the weight of the secondary cage, mirror, and
electronics
Arod
= the cross sectional area of a single rod
E = the elastic modulus (Young's modulus) of the rod
steel
a1
= the angle that the lower rods make with respect to the telescope
radial direction (i.e. with respect to horizontal when the telescope
is at the zenith)
a2
= the angle that the upper rods make with respect to the telescope
radial direction