# How do I calculate the ideal travel distance of the ball screw per revolution of the stepper or per step for the following hardware?

Question How do I calculate the ideal travel distance of the ball screw per revolution of the stepper / per step for the following hardware (I believe this is called the lead)?

Background I have a pre-built 100mm Ball Screw Drive Linear Slide purchased from Amazon.

The specifications of this assembly are as follows.

• Material: Aluminum

• Boundary dimension: 40mm

• Shaft diameter: 12mm / 0.47inch

• Accuracy: 0.01mm

• Effective stroke: 100mm / 3.9inch

• Ball screw length: 165mm

• Sliding block: 60 x 80 x 30mm / 2.36 x 3.15 x 1.18inch

• Pitch-row width: 68mm (max)

• Pitch-row length: 60mm

• Screw: M6

• Slide table size: 190 x 40 x 20mm / 7.48 x 1.57 x 0.79inch

• Weight: Approx. 1432g

The stepper motor included with this module is 42BYGH48.

Looks very similar to the ones defined here with the same wire out colours. Looking into this I found multiple sources refer the desired number this as the lead (lead is the linear distance traveled by the nut or screw after one full rotation). Page 2 of this source talks about this but I don't really follow where these numbers have come from.

Can anyone suggest a good source to help explain to me what I need to calculate this for myself? What values am I missing from my setup to calculate this?

Sorry, I suspect I'm thinking about this all wrong and look like an idiot! had a long day at work and just spent an hour digging around and made no progress! I was certain someone here could point me to exactly what I needed!

• Lead is basicaly the pitch of the screw. Commented Mar 16, 2021 at 6:13
• assuming a single-start screw (which it ought to be), 1 unit of pitch per revolution. Commented Mar 19, 2021 at 17:48

Screw: M6

M6 coarse has a 1 mm pitch. M6 fine has a 0.75 mm pitch. Unless otherwise stated assume coarse, so 1 mm pitch.

Your motor is 1.8° per step. This is 360/1.8 = 200 steps/rev.

The screw will advance 1 mm every 200 steps.

As usual with Amazon, there are no datasheets and poor specifications in the ad which is what you linked to. There's a question in the comments:

Question: What is the distance traveled per 360° rotation of the stepper motor? (# of threads p/mm could help). Answer: 4 mm per rotation. this is a solid bit of kit By andrew m. on 16 March 2021

So the slide will advance 4 mm every 360° or 200 steps.

Usual advice applies: No datasheet? No sale!

• Isn’t the OP asking about ball screws? Commented Mar 16, 2021 at 14:23
• See the update, thanks, @Eric. Commented Mar 16, 2021 at 14:32
• Thanks for the comment, you found my question to the supplier on the amazon listing I want to verify what he has said at some point this weekend. I do plan to build this rig from the ground up eventually but to get the project started I wanted something pre-built, immediately regretted it when I sat down and tried to think about how to make it work! Commented Mar 17, 2021 at 10:41

In general the ideal travel per 1 revolution of the motor, will be depended on two things:

## screw dimensions

the screw dimensions (in your case M6) and more specifically the pitch (see image below)

Although its usually forgotten ISO 261 defines a coarse and a fine pitch. See wikipedia. The pitch should be written on the designation of the bolt M6-1.00 (coarse) or M6-0.75 (fine). If its omitted then its assumed to be the coarse.

So, technically, for one rotation of the shaft(/screw) you get a travel displacement of 1 pitch. (this can get complicated when you have a multi-start screw but I won't get into that).

## Gear ratio

If you notice in the previous section I highlighted one revolution of the shaft(/screw). The reason is that (instead of directly coupling the shaft to the motor) you might intervene some sort of gearbox, or pulley or chain.

In that case, one rotation of the motor results in a multiple of rotations of the shafts. The multiple is defined as gear ratio, and it has multiple definitions. The most basic one is:

$$i = \frac{n_{out}}{n_{in}}$$

where:

• $$i$$: Gear ratio
• $$n_{out}$$: rpm at output (e.g. shaft)
• $$n_{in}$$: rpm at input (e.g. motor)

## bottom line

So 1 rotation of the motor, would result in 1 rotation of the shaft. Or equivalently

$$\text{distance travelled} = i \cdot \text{pitch}$$

• Thanks that explains it really clearly, and matches what I've seen in my testing Commented Mar 17, 2021 at 10:38