Balancing an equatorial

Balancing an equatorial

In any equatorially mounted telescope there are three axes about which
moments have to be determined. It is important which order these are
considered. They are as follows:
i)    radial equilibrium about the tube axis
ii)   longitudinal equilibrium about the declination axis
iii)  declination equilibrium about the hour axis

If the telescope tube is not in radial equilibrium it will not be possible
to adjust the system cg to the intersection of the declination and hour
axes, for all possible attitudes of the tube assembly. Radial balance is
the most awkward to achieve, and is crucial. Radial out-of-balance is the
most common cause of the complaint, "My telescope will not balance in all
positions."

To determine the radial momenta about the tube axis, measure the offset of
all the non-symmetric tube furniture (i.e. rackmount, finder, guide 'scope,
accessory mounting brackets &c). Ignore symmetric items, i.e. the primary
and secondary mirrors & cells, the spider, cradle rings &c. Weigh, or
calculate the weight of each item. Adopt an arbitary reference plane,
transverse to the tube's mechanical axis. A useful reference plane might be
in the direction of gravity when the tube is horizontal, pointing due south.
Call this plane X-X'. Prepare a scale diagram depicting the tube profile,
and each weighed item. Measure or estimate the angular distance of each
item, anti-clockwise, looking down the front of the tube from the reference
plane.

The radial momenta of each item will then be:

Moment = radial offset x weight x cos(angular distance from X-X')

Always measure the angular distance from the same reference, using 360°
notation, in the same anti-clockwise direction. Repeat the procedure using a
reference plane rotated 90° clockwise. Call this plane Y-Y'. Sum the momenta
about each plane.

As an example, I performed the calculation for my 10-inch f/10.6 Calver, with
the following results:

Moments about X-X'

rackmount = 12" x 7lbsf x cos270° = 0inlbs
lower counterwights = 8".75 x 56.68lbsf x cos270° = 0inlbs
Cooke 2-inch finder = 12" x 7lbsf x cos330° = 73inlbs
mounting bracket + 400mm lens = 9".75 x 11.8lbsf x cos0° = 115inlbs
mounting bracket + 150mm lens = 9".75 x 10.8lbsf x cos180° = -105inlbs
upper counterweight = 8".75 x 22.42lbsf x cos90° = 0inlbs
4-inch Guide 'scope = 10".25 x 7lbsf x cos135° = -51inlbs
Ottway 2-inch finder = 9".25 x 2.625lbsf x cos210° = -21inlbs

Moments about Y-Y'

rackmount = 12" x 7lbsf x cos0° = 84inlbs
lower counterwights = 8".75 x 56.68lbsf x cos0° = 496inlbs
Cooke 2-inch finder = 12" x 7lbsf x cos60° = 42inlbs
mounting bracket + 400mm lens = 9".75 x 11.8lbsf x cos90° = 0inlbs
mounting bracket + 150mm lens = 9".75 x 10.8lbsf x cos270° = -0inlbs
upper counterweight = 8".75 x 22.42lbsf x cos180° = -196inlbs
4-inch Guide 'scope = 10".25 x 7lbsf x cos225° = -51inlbs
Ottway 2-inch finder = 9".25 x 2.625lbsf x cos300° = 12inlbs

Sum of momenta about X-X' = 11inlbs
Sum of momenta about Y-Y' = 387inlbs

To calculate the weight (W) and orientation (A°) of a counterpoise necessary
to bring the tube into radial equilibrium, we must adopt a suitable radial
offset for the counterpoise (e.g. r = 8".75) and solve the following
simultaneous equation:

i)    8".75 x W x cosA + 11 = 0
ii)   8".75 x W x cos(A+90°) + 387 =0

divide (ii) by (i)    (cos(A+90°) = -sinA)

hence  8.75.W.sinA/8.75.W.cosA = tanA = 387/-11 = -35.182
(sinA positive - 2nd quadrant - recall all trig ratios +1st quad; sin +2nd; tan +3rd; cos +4th)
therefore A = 91°.63

subst. A in ii) hence 9.75.Wcos181°.63 = -387
& solving for W                       W = 44.25lbsf

To check the calculation, determine the resultant sum of the radial momenta:
about X-X' = 8.75x44.25cos91°.63 + 11 = 0inlbs
about Y-Y' = 8.75x44.25cos181°.63 + 387 = 0inlbs

Where the counterpoise is located along the length of the tube is arbitary.
It is its orientation with respect to X-X' which is important. I decided to
split the counterpoise into two equal weights, placed either side of the
dec. axis.

Once you have determined the radial equilibrium, calculating mometa about
the declination and hour axes is a matter of simply weight x axial distance.

This page was created by SimpleText2Html 1.0.2 on 30-Jan-102.


 

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