Torque calculation

I have a physics problem that I haven't been able to find a good solution for on Google.

I'm designing an astronomical binocular mount that will permit extended periods of hands free observing. It will use a modified color camera board and an Arduino Uno to track objects in the sky. The MCU, via a servo control board, drives two high-torque servos which are attached to the altitude and azimuth axes of a pair of gimbals. They operate in either auto-track mode or in manual mode via a hand controller. The mount itself is lightweight tubular aluminum, but the binoculars themselves are quite heavy. Although they are very heavy, their CG is located precisely at the center of each gimbal so the forces acting on the servos should always be the same regardless of position. The pivot points of the gimbals consist of lubricated ball bearings so the resistance to movement as a result of friction is very low.

I'm trying to figure out how powerful the servos need to be in terms of torque, in order to move the binoculars around. The binoculars weigh 10lbs. The maximum angular speed at which the binoculars will ever be moved is 5 degree/second. The length of the each servo arm is one inch. The length of the lever arm connecting the push rod to each gimbal is also one inch so there's a one-to-one ratio of servo to gimbal movement. The length of the binoculars are about 15 inches, but they're not mounted at the center of their length due to the fact that the objective end is heavier than the eyepiece end, but for the sake of argument, lets say they're mounted halfway along their length, both vertically and horizontally.

I'm not sure if there are any other details needed for the calculation. If not, what is the correct equation to solve for the torque?

If the device is perfectly balanced as you suggest, then the torque needs to be sufficient to overcome the static friction in the device, and sufficient to provide the acceleration you require after subtracting the torque needed to overcome the dynamic friction. So when moving at a steady 5 degrees/sec, you need only enough torque to overcome the dynamic friction.

The friction can't really be calculated, only measured.

If you know how fast you want to accelerate up to 5 degrees/sec then you can calculate how much torque you need to provide this acceleration, but you first need to calculate the moment of inertia of the assembly. Google for "moment of inertia" for more on this topic.

Unless you are looking to get fast acceleration, then it sounds to me that you need very little torque.

Thanks, dc. I'll check out moment of inertia. I'd say 2 seconds is a reasonable amount of time to expect it to get up to full speed. I expect the static friction to be very low, due to the quality bearings on each axis.

I suggest viewing this video from a guy in my robotics club, it's very useful and interesting.

Not only interesting but totally strange to boot. He mixes engineering units with gay abandon and ends up with gear ratios being specified as a result of some mathematical enigma derived from sliding a mass down an incline with no relevance to the power of the drive motor. Certainly not Mechanics of Machines as taught when I was at uni. Perhaps the publishing date was 1st April

But what the hell, if it works for you then .......