How do determine proper gear size?

[BiotechFusion]

I never did much with gears. I have a rack with 20 teeth and I want a very small gear to fit on it. I would think that if I wanted the size of the gear to be, say, 1/2 the length of the rack in circumference, then the gear would have 1/2 the number of spurs but with the same length and height as the teeth on the rack, but I don' know for sure. I mean you can't just have different sizes and have the gear with the same number of teeth on each gear could you? At some point it seems like it wouldn't fit.

Edited April 21, 2016 by BiotechFusion

[John Cuthber]

You are right; you can't just "choose" the gear size.

[BiotechFusion]

You are right; you can't just "choose" the gear size.

Do you mind expanding on that and answering the question? For some reason I can't find a single source that just says "if the gear is this size, use this many spurs of this size..." or "if you want this many spurs, use a gear of this size..."

Edited April 21, 2016 by BiotechFusion

[Danijel Gorupec]

To perfectly match gears their 'pitch' value must match. When you have a rack, it is easy to measure its pitch distance because it equals to the distance between two teeth (like between the center of one teeth and the center of the next teeth). With gears (pinions) that have small numbers of teeth it is not so easy to measure their pitch distance...

 

[As you noticed, gears pitch do not have to match perfectly, and gears will still mesh. However in this case you might have problems with wear, noise, lash, vibration...]

 

In practice, instead of 'pitch' more often the value called 'module' is used. The module is equal to pitch/pi... To very roughly calculate gear diameter you can multiply its module by number of teeth (the actual diameter is somewhat larger than that).

 

Generally, the minimum number of teeth a gear can have is about 17, most people say. To have smaller number of teeth, some additional precautions are needed (come with price of increased wear or decreased strength...)

 

There are standard modules for imperial and metric gears, and it is likely that your rack is made using some standard module.

[John Cuthber]

Ignore the teeth for a minute and consider a rubber wheel running on a "rack" with a smooth surface.

 

You say " I would think that if I wanted the size of the gear to be, say, 1/2 the length of the rack in circumference, "

Yes, you can choose a "gear" that size. you can calculate the size of wheel whose circumference is half the length of the rack.

 

Now imagine you cut teeth into the rack and the gear.

We can start with the idea of lots of teeth and a finely toothed rack.But if there are a thousand teeth on the rack and you want them to match up to the gear then the gear has to have 500 teeth- and it still needs to have the same circumference (strictly- the same pitch circle) as before.

 

If you have a gear with twice the circumference then, for the teeth to still match up with the rack, you will need twice as many teeth as before.

But for each turn of the wheel it will then move 1000 teeth along the rack- so that' no longer the "half way" you specified in the first place.

 

So. let's go back to the original question.

 

" I have a rack with 20 teeth and I want a very small gear to fit on it. I would think that if I wanted the size of the gear to be, say, 1/2 the length of the rack in circumference, then the gear would have 1/2 the number of spurs "

Well, if the rack is some length L then the pitch of the teeth must be L/20

And if you want the gear to have a circumference of half the rack length then that makes the diameter 1/2 L /pi

That's it- it's a fixed size. You can't choose what size you want

And - to get them to meet the teeth on the rack, the teeth on the gear will also have to have a pitch of L/20

What you can change is how many turns of teh gear you get for the length of the rack.

[BiotechFusion]

Hmm alright that's some good info. So I guess what you're saying is if I want a smaller gear, the teeth need to be the same size as they are on the rack and I need to have more teeth so that I can make the teeth on the gear smaller. But, there simply must be some way to have a few difference size options in the gear for a rack with a set number of teeth, I still don't see a clear formula for that. Let's say a rack is 1 meter/foot in length and 5/12 of a meter/foot in height, and the housing for the entire rack and pinion has to be a max of 12/12. So, the gear can only have a maximum of 7/12. How do I calculate the number of teeth on a gear of a few sizes between 0 and 7/12 of a unit?

 

Also what's the math like for slanted gears? I don't really know why slanted gears exist cause they seem kind of weaker and stubby, but I was just thinking that if I wanted to save on the size of a gear when height/length is a limitation and reduce the height of the spurs from the gear, I could obtain the same surface area of contact by having slanted teeth interlock, but I also don't know any limit for how big a tooth can be cause at some point it would probably break off.

[Danijel Gorupec]

Let's say a rack is 1 meter/foot in length and 5/12 of a meter/foot in height, and the housing for the entire rack and pinion has to be a max of 12/12. So, the gear can only have a maximum of 7/12. How do I calculate the number of teeth on a gear of a few sizes between 0 and 7/12 of a unit?

 

If you know the pitch (tooth distance) of your rack, then you can roughly estimate:

- diameter of the pinion in dependence of number of teeth: d=p*N/pi

- number of teeth in dependence of the pinion diameter: N=pi*d/p

(where 'd' is pinion diameter, 'N' is number of pinion teeth, 'p' is the pitch of your rack, and 'pi' is 3.14...)

 

But do not forget that in practice the number of pinion teeth cannot be very small. I don't think it can even work for N<10, and even this with serious disadvantages. Only with N>17 you can hope for a nice, reliable and cheap solution.... As you concluded, if you need a smaller diameter pinion but you cannot decrease number of teeth any more, the only way would be to use finer (smaller) teeth on both, rack and pinion.

 

I don't know what does 'slanted gear' mean in English... it can mean helical or bevel gear, I suppose. Helical gears are used the same way as spur gears (for parallel axes) but will work smoother (less vibrations and noise) and will cost more. Bevel gears are used when gear axes are not parallel.

[BiotechFusion]

Oh, yeah I guess it is that simple, I guess I just didn't know what to do because...do I need the height of each tooth to be the same as well? So no matter what size of rack or pinion, all teeth need to have exactly the same dimensions?

[Enthalpy]

Gears are a matter for specialized companies. Designing a gear that lasts for hours is a matter of experts. Machining a gear takes special tools (available from tool producers, sure). Then, you need special materials and heat or surface treatments. Even at a mechanical engineering company, designers wouldn't draw a gear, but instead specify the parts to be bought, possibly after advice from the specialized company.

 

Gears are standardized. They are grouped in compatible families through their "module", which isn't exactly a pitch; the module comes in standardized series and must match for long and smooth operation.

https://en.wikipedia.org/wiki/Gear

Once chosen, the module defines completely the tooth profile and, with the number of teeth, the wheel diameter.

 

So to my opinion, you're seeking too much detail, since a good gear is both bought and standard. All you must define is a module (tooth size) and the number of teeth at the wheels. Note that the numbers of teeth are normally chose mutually prime to reduce the wear, so equal numbers would be avoided.

 

A reasonable approach would be to seek an online gear catalogue to check that existing gears transmit the torque, speed, power you want.