I think it's about time to talk about lead screws. So, what is this lead screw that is always mentioned throughout this website. Lead screws also have all of these terminologies and esoteric language that I will demystify. Maybe you will also start to understand the math behind the linear motion and how different lead screws will provide various linear motion characteristics.
Essentially, lead screws are simply screws that can provide linear motion when a turning motion is applied. Lead is defined as the helix that is common to all screws. A good example of a helix is the DNA molecular structure that exhibits a double helix. In other words, the thread that spirals around the screw.
The thread on a standard screw has only one groove that spirals around the shaft that pairs with a nut that reflects the same thread within the inside of the wall of the bore (hole). Different size screws have different leads, or threads per inch. In most cases, even similar sized screws have different threads per inch. Let's say, we have a couple of #10 screws. If the screw has coarse thread, if is considered as having fewer threads per inch than a fine thread screw. A #10 fine screw would have 32 threads per inch, in contrast to a coarse thread having 24 threads per inch.
So what is threads per inch anyway... You will sometimes see this represented as TPI. TPI can also be referred to as turns per inch, and I will get into that explanation later (it's necessary, trust, me). Threads per inch is the number of grooves that you can count in one linear inch. Get out that ruler you've been hanging on to since elementary school and hold it next to a screw. You will want to count the groove/thread pairs within the inch. You can also find this TPI information on the screw packaging. You might have seen a reference like 1/2"-13 X 2": (1/2") is the diameter of the screw, (13) is the threads per inch or TPI and (2") is the length of the screw. In the case of these "single start" screws, turns per inch is the same as threads per inch. That is to say, if you turn the lead screw, say 13 times for the above example, and there is a nut on the lead screw that you are holding so it doesn't turn, the nut will be almost exactly one inch from where it was prior to turning the screw.
Yep, I said it, single start! There is also a characteristic with these screws that determines how many threads there are spiraling around the shaft of the screw. A screw with multiple starts has more than one groove spiraling around the screw. the above DNA double helix explanation, if applied to a screw would be called a 2 start screw, since there are two threads spiraling around the screw shaft. I need you to put on your mind's eye for a moment. Suppose you have a spring in your hand... now imagine with your minds eye (if this is difficult for you, take apart your ball point pen and grab that spring at the tip) that you stretch out that spring. Now there is a bunch of space between the spiral. A screw needs its threads pretty close together. Manufacturers have developed lead screws to vary the rate at which linear motion occurs. There are a couple of ways to do this: 1. use a screw with a single start and find a motor that can turn very fast, or 2. use a motor with good torque and use a lead screw that exhibits a very low "turns per inch" count.
Now, back to my minds eye explanation... If the threads are stretched out so the threads per inch measure, say 4, there would be an awful lot of space between each groove. This is a problem with screws and nuts, especially if the nut is not very long. A way to establish great contact between the lead screw and nut would be to add more grooves. An example: let's say you have a 1/2"-8 X 6' with 2 starts. In the previous example, you know that the 1/2" is the diameter and the 8 is the threads per inch, but the difference here is the number of starts. The actual "turns per inch" is actually 4, not 8. This is a double helix, wound in a much tighter configuration than the DNA. All of the threads are next to each other, measuring exactly 8 threads in one inch. These are typically found in ACME types of precision lead screws. To find the turns/inch, simply divide the TPI with the the number of starts. So, 8 TPI / 2 starts is 4 turns per inch. Ok, so let' say we have this type of screw: 1/2"-.500. Ahhh... A trick question. This would be considered a high-lead screw, which is the same as saying this screw has say 8 TPI and 4 starts. To find the lead, simply divide the starts over the TPI: 4 starts / 8 TPI is .5 lead. I mention this because you will find this terminology with some manufacturers.
How does this translate to speed? Easy... If a motor turns 2 times for every second, then a screw that exhibits 2 turns per inch will travel faster than a screw that has 10 turns per inch. In other words, the 2 turns per inch will get to one inch in one second; whereas, the the 10 turns per inch screw will get to one inch in 5 seconds. Get it? 2 turns/second X 10 turns/inch = 5 seconds/inch. The mathies out there can do your unit calculation by taking the reciprocal of turns/second and multiplying it to turns/inch and cancel out the turns, so you are left with seconds/inch.
Now comes the easy math... How do you calculate the number of steps per inch. That's also easy. This will involve something I haven't yet explained (I have elsewhere in this website), the motor. Stepper motors have steps. Steps per revolution to be exact (I will refer to this as steps/turn). A common number of steps/turn is 200, which means that for each time a motor moves to the next step, it rotated 1.8 degrees. This explanation would get long trying to explain the steps in a stepper motor, so for this example, just remember that stepping motors turn from one step to another by receiving pulses to a combination of its coils (magnetized pieces in the motor when energized).
So you take the number of steps/turn and multiply that with the turns per inch (not threads per inch). That's it! I told you it was simple. An example, 200 steps/turn x 10 turns/inch yields 2000 steps/inch. Remember, those turns cancel out. Wow!! Now that's precision. Well, yes and no, depending on the overall mechanical system and backlash reduction. Generally, the precision will be increased with the increase of the turns/inch. If you have only 2 turns/inch... sure it'll be fast, but the precision will be reduced to 400 steps/inch. Still pretty good for woodworking, but I wouldn't recommend PCB (printed circuit board) routing. That's where you will need to balance, or determine the necessary turn/inch and velocity you need for your application.