[ATM] Designing a simple fork with desirable flexure characteristics

["Tom Krajci" <tom_krajci@tularosa.net>]

I plan to change my German equatorial mount to a fork.  It will be used for
time-series photometry runs that can last all night.  Accurate pointing and
polar alignment are highly desirable.  I want the scope/mount to flex in
small, but predictable ways so that I do not need a complex software
solution to improve pointing.

The fork will deflect under the load of its own weight, and that of the
telescope/CCD.  This will make the effective polar axis lower, which has an
impact on polar alignment and pointing accuracy.

Possible solution:  elevate the polar axis slightly to offset this droop.
Is that all we have to worry about?  Unfortunately, no.

One problem with fork designs is that they (if not designed properly) will
flex/droop more in some orientations than others.  This spoils polar
alignment and pointing accuracy.  This means I must design the fork to have
similar resistance to droop (stiffness) in various orientations.  See:
http://www.dfmengineering.com/news_eng_article_3.html for a discussion of
the problem.

In some orientations the fork is only subject to moments (bending forces).
This happens when the telescope is pointing at an hour angle of +/- 6
hours - looking at the horizon.

In some orientations the fork is subject to both moments (bending forces)
and torques (twisting forces).  This happens when the telescope is pointing
at an hour angle of 0 hours - along the meridian.

OK, then I need to analyze the design to make sure stiffness is the same in
different orientations.

Design constraints:
- Steel is the material of choice
- Square tubing, ? inch wall thickness will be used.  Available
sizes/choices are 2, 2 ?, and 3 inches wide.
- If the crossbar needs to be made from a thicker section of square tubing
than the fork arms, that is not a problem
- If crossbar length, or fork arm length need to be changed slightly to
achieve the desired stiffness equality at various orientations, that is
acceptable
- Fork length is 22 inches (from crossbar axis to declination axis)
- Fork width is 18 inches (from one fork arm axis to the fork arm other
axis)
- Telescope/load will be 50 pounds

For bending/moments the moment of inertia (I) for a square tube is:
 a^4/12 - b^4/12 where a is outer width of the square tubing and b is inner
width
See: http://www.willbell.com/tm/dobtel.htm
page 55 (fig. 3.5)

For twisting/torsion the polar moment of inertia (J) of a square tube is:
a^4/6 - b^4/6 where a is outer width and b is inner width.
See:  http://www.efunda.com/math/areas/rectangle.cfm
specifically the value for "Polar Moment of Inertia about the Zc axis" and
set b and h equal to each other, and simplify the resulting expression.

I mention these formulae because I'm using some finite element analysis
software, Grape, to perform the analysis.  http://www.grapesoftware.mb.ca/
But I don't want to blindly accept the software output.  I'll feel better
about the answers they provide if I can manually work some simple beam
bending/twisting problems.and the answers are in reasonable agreement.

Also, as pointed out by Barry Jensen, this set of tables has pre-calculated
values for I and J.which will differ slightly from the simple equations
above because real square tubing has rounded corners.
http://www.steeltubeinstitute.org/pdf/brochures/dimension_brochure.pdf

Let's start with a 'baseline' fork, and then change some parameters to see
how we may be able to 'tune' its stiffness at different orientations.  Using
Grape software we'll use 2 inch square tubing, ? inch wall.  The cross bar
will be 18 inches and the fork arms will be 22 inches.

Assumption:  because of symmetry, we only need to depict one fork arm and
one half of the cross bar to perform the analysis.

We'll analyze angular deflection of the end of the fork arm (polar axis
droop) at two orientations
- oriented to look at the meridian (hour angle 0)
- oriented to look at the horizon (hour angle 6)

We'll use a latitude of 33 degrees to orient the force of gravity and
telescope payload with respect to the fork.  To keep it simple in Grape, the
fork will be built to 'stand up' in the +Y direction and be 'wide' in the +X
direction.  This means in Grape that we need to apply gravity and load
components in two axes so the resultants are at an angle of 33 degrees.
Using some trigonometry, the two components will be as follows:
- always in the -Y direction:  0.545 (directed 'down' the fork's polar axis,
which is not depicted in our Grape simulation)
- either in the +X or -Z direction:  0.839 (this is the force that does the
most flexing/distorting of the fork that is of concern to us, and its
direction depends on whether the fork is oriented toward the meridian or
horizon).

Our telescope payload is 50 pounds total, or 25 pounds per fork arm.  When
this force is resolved into two axes, we get 13.6 pounds in the -Y direction
and 21.0 pounds in the +X or -Z direction.

Grape FEA software tells us both the linear displacement of a given point on
the fork, and its angular displacement as well.

Baseline "light" fork (2 inch square tubing, ? inch wall.  Cross bar 18in,
fork arm 22in.)
- Meridian:  5.56e-4 radians about the X axis, which is 1.91 arcminutes
- Horizon:  4.43e-4 radians about the Z axis, which is 1.52 arcminutes

Difference in polar axis droop is 23 arc seconds, and the fork droops more
when looking at the meridian.  I would prefer the difference in droop to be
a lower value.

Let's change the fork by making the cross bar of 2.5 inch square tubing, ?
inch wall.

Modified "light" fork #1 (Cross bar is now 2.5 inch square tube.)
- Meridian:  3.37e-4 radians about the X axis, which is 1.16 arcminutes
- Horizon:  3.26e-4 radians about the Z axis, which is 1.12 arcminutes

Difference in polar axis droop is 2 arc seconds, and the fork droops more
when looking at the meridian.

At this point it looks like I've found an acceptable fork design in terms of
polar axis droop at various orientation, however let's examine the parameter
space a bit more.  Let's go back to the baseline fork design where all
components are 2 inch square tube, but make the cross bar a bit longer.

Modified "light" fork #2 (Cross bar is still 2 inch square tube, but 2
inches longer.)
- Meridian:  6.31e-4 radians about the X axis, which is 2.17 arcminutes
- Horizon:  5.01e-4 radians about the Z axis, which is 1.72 arcminutes
(Note:  in our analysis we only look at half the cross bar, so overall the
fork is now 4 inches wider.)

Difference in polar axis droop is 27 arc seconds, and the fork droops more
when looking at the meridian.  (Note:  We did not change the droop behavior
very much this way, and we made it less desirable.)

We'll change one more parameter compared to the baseline design - fork arm
length.  Let's make it a bit longer.

Modified "light" fork #3 (Everything is 2 inch square tube, but fork arm is
2 inches longer.)
- Meridian:  6.38e-4 radians about the X axis, which is 2.19 arcminutes
- Horizon:  5.09e-4 radians about the Z axis, which is 1.75 arcminutes

Difference in polar axis droop is 27 arc seconds, and the fork droops more
when looking at the meridian.  (Note:  Again, we did not change the droop
behavior very much this way, and again made it less desirable.)

Based on the above results I will make my fork with 2 ? inch square tube for
the cross arm, and 2 inch square tube for the arms.


In this analysis I did not take into account any potential increase in fork
stiffness (especially when pointed at the horizon) from the telescope/cradle
being attached to both fork arms.  I assume this will not make a significant
change in the analysis.

I only changed the width of square tubing, and length of cross bar and fork
arm.  If you need other parameters to change to 'tune' your fork design, you
can also change the cross section of the cross bar and fork arms.  In other
words they don't have to be the same width in both directions; you could use
rectangular cross sections.

I hope this helps.

Please let me know if I have made any significant errors in this analysis or
assumptions.

Tom Krajci
Cloudcroft, New Mexico
http://overton2.tamu.edu/aset/krajci/



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