# How to calculate the steps needed for a motor to rotate a circular object in a pulley configuration?

I have a stepper motor with 200 steps per revolution and a pulleys configuration as following: 3 pulleys with 100 mm diameter each on 1184 mm PCD and are equally spaced (120 degrees) from each other. A large ring with a diameter of 1074 mm is placed on the center on the pulleys configuration (center of ring to center of each pulley is 592 mm). A belt of 3626.5 mm length and 30 mm width is positioned around the pulleys and contacts the ring. The motor is installed vertically on the bottom of one of the pulleys (concentric on the same axis). I want to know how many steps it takes for the motor in order to have a full revolution on the large ring.

Here is a simple drawing of the configuration

• Are the pulleys and the large ring in contact? I would have expected the 592mm to read 587mm and the PCD to 1174mm, not 1184mm. May 19, 2015 at 8:18
• Your description is confusing, and the drawing doesn't clear up everything. I don't see where the motor is connected. There is no belt shown. What does it mean for the motor to be installed "vertically"? "On the bottom of one of the pulleys", huh? This question needs to be closed in its present form. We do engineering here, not hand waving. May 19, 2015 at 11:31
• Hi Pulley, welcome to engineering.SE. As Olin has mentioned, your question is unclear in it's present form. The best way to make it clear is to add the motor and belt to your drawing since text descriptions can be hard to follow. Can you please update your question by editing it? May 19, 2015 at 13:05
• ...I don't get it why people keep insisting on "unclear". The motor drives one of the pulleys directly as explained in the text, so the ratio of rotation is 1:1 - it doesn't matter which pulley. The way the belt goes is totally moot because at worst it would affect direction of rotation; you can't change the gear ratio without using more than one belt. The information provided is perfectly sufficient for the purpose of answering this.
– SF.
May 20, 2015 at 9:44
• Thank you everyone for you informative feedback. I've attached an additional image describing the way the belt is installed and beneath the pulley on the right is the motor. May 21, 2015 at 9:26

You provided a lot of redundant information which is moot to the answer. Whether the pulleys drive the ring through contact, the belt, how the belt goes etc is all moot to the answer. The only thing that matters is the gear ratio between a pulley and the ring (since the motor drives the pulley directly, the gear ratio between the motor and the pulley is 1.) The only thing that could change is direction of rotation - if the belt makes "8" style double loop or the transmission is through contact and not belt, it would rotate the ring in the opposite direction.

On one full turn of the pulley, the belt travels the distance of its circumference. The same distance of travel is covered by any point on the circumference of the ring. Since circumferences are directly proportional to radii and diameters, we can compare these for the gear ratio instead; the ratio will be the same.

$$1074mm/100mm = 10.74$$

That means for 10.74 rotations of the pulley the ring makes one turn.

Now simply multiply this by number of steps of the motor per turn of the pulley:

$$10.74*200 steps = 2148 steps$$

The result will be slightly different if you take into account some grooves, teeth, slippage, backslash, stretching of the belt and other factors, but the idea remains the same - the distance of travel of the belt is the distance of travel of the point where the belt makes efficient contact with a wheel. The ratio of circumferences of the circles drawn by these points is your gear ratio; times $n_{steps}$ you get the number of steps.

• Thank you SF for the clarification. Regarding the redundant info, it was so that I get everything clear in case I didn't convey it well. However, I am not sure which one should be more accurate this answer or the answer provided by am304 taking in consideration the gap between the pulley and the ring. May 21, 2015 at 9:31
• What transfers the movement? If there's no contact between the pulleys and the ring, the belt transfers the movement and in this case the "2148 steps" is correct - whether there's 5mm distance of 5000 is moot, changing backslash at worst - it's the gear ratio between pulleys and the ring only. If they are in contact and providing movement while the belt slips over the surface merely stabilizing the contraption, the latter is correct.
– SF.
May 22, 2015 at 12:47
• I get it now. So in my case since the belt is what drive the ring and there is no contact between it and the pulley, the ratio is derived from their exact sizes and should neglect the spacing between them. Thank you SF. May 24, 2015 at 11:52

Assuming a perfect transmission, the amount the pulley needs to turn for a full revolution of the large ring is governed by the "gear ratio":

$\theta_{ring} = 2 \pi * 1074 / 100 = 67.48$ rad or 10.74 rev.

Assuming you are full stepping, this is 2,148 steps. If you are half-stepping, it's 4,296 half-steps or if quarter-stepping, 8,592 quarter-steps.

In practice, it's possible that teeth are jumped, depending on how well the mesh is between the pulleys and the belt, and the belt and the ring, in which case you'll get some slipping and won't get quite a full revolution.

EDIT

It looks like there is a 5mm radial gap between the ring and the pulleys, so I would adjust the calculations as follows:

$R_{ring} = 1074/2 = 537$ mm

$R_{pulley} = 50$ mm

$\theta_{ring} = 2 \pi * (537+2.5) / (50+2.5) = 67.57$ rad or 10.276 rev.

Assuming you are full stepping, this is 2,055 steps. If you are half-stepping, it's 4,111 half-steps or if quarter-stepping, 8,221 quarter-steps.

• See my comment, those calculations might be slightly inaccurate if I misunderstood your question. May 19, 2015 at 8:57
• Thank you for your explanation am304 ; that gave me a clue to how and from where to start the calculations. May 21, 2015 at 6:46
• @Pulley See my edited answer to account for the radial gap between the pulley and the ring. If this helped answer your question, please consider accepting the answer so that other users with similar issues can more easily find a solution to their problem. May 21, 2015 at 8:33
• Thanks a lot for your efforts in providing the correct answers. So should I take the gap into account for the calculations? May 21, 2015 at 9:40
• @Pulley Yes, I think so. It's almost as if you had gears of radii 539.5 and 52.5mm in contact. May 21, 2015 at 9:50