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I am not a mechanical engineer, so just trying to figure this out. I want to attach a servo to my telescope focus, but the servo only has about 180 degrees motion, whereas you need to turn the knob about 540 degrees for the full range of focus. It has a rack and pinion gear, so my plan was to find a pinion gear with more teeth (maybe 48?) to get the full range in 180 degrees.

The pinion gear has 16 teeth and an outer diameter of about 8mm. I measured 10 teeth on the rack which was about 15mm. The diameter of the shaft is about 5mm. My question is, what size gear would I need, how many teeth, what pitch? Does it exists? My best guess is 48 teeth would be needed, but I am not sure about my pitch. Also, if the gear exists, do they make them to attach directly to a servo? Or will I need to get a shaft and a shaft to servo coupler?

Thanks

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    $\begingroup$ I agree a stepper motor would probably be a better solution here, but if you really want to change the pinion diameter you need a pinion with the tooth pitch and tooth profile designed to match the rack. If you have 3 times as many teeth, the diameter of the pinion will be approximately 3 times as big, but "approximately" won't be good enough to design a reliable system. You may be able to get the specification of the rack from the scope manufacturer, or an astronomy forum. Assuming the rack is a standard part, it should be possible to use it with a range of different "standard" pinions. $\endgroup$ – alephzero Dec 4 '16 at 1:01
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I suggest that you use a stepper motor, not a servo for this application.

Steppers can be controlled in small rotational increments. The speed of rotation and the torque delivered by the stepper motor are easily determined and controlled. Furthermore, a stepper with an encoder provides an accurate and repeatable method of precisely positioning a system.

Given your gearing, the final answer depends on several factors with the torque and speed driving the design. From the appearance of the mechanism a small NEMA 17 motor with 1.8 degree steps would be more than sufficient assuming that the mechanism is reasonably free-running. If it is stiff, a larger motor with a larger current supply would improve performance.

Stepper controllers are quite simple and easily available. For a telescope, a LiPo battery would make sense so I would consider a NEMA 17, 3S LiPo and a small controller with an external potentiometer/PWM used for servo testing to adjust the position, to be sufficient

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