Mechanics, gear ratio problem

[GlennR]

For a school project I have drive something with a ratio of 2. How I accomplish this is completely up to me. I chose to do it with two sets of gears, as shown in the picture below. I tried to make it as clear as possible how the motion goes and which gear drives which. Now my problem is more mathematical since I am trying to figure out which single ratios are possible that equal 2 together (since there are two sets of gears) whereafter I can decide which are most suitable/optimal for my design.

With some logical thinking I know that the following ratios are possible:
[MATH]1/1 * 2/1[/MATH][MATH]4/5 * 5/2[/MATH][MATH]8/5 * 5/4[/MATH][MATH]4/3 * 3/2[/MATH]
However, I am looking for a more mathematical approach in figuring out if there are more gear ratio's suitable. To not have an endless quantity of data, 1/6 is the outer extreme.

 

[Dr.Peterson]

What would make one set of gears better than another? You can list many, many possibilities (e.g. 26:19 * 19:13), but before doing that I would want to think about what your goal will be (so you can focus on listing the useful ones).

In your drawing, it looks like you want the sum of the radii of each pair of gears to be the same (and therefore, if teeth are all the same size, the total number of gears in each pair would be the same). Is that a goal? If so, then maybe you need to think not just about the (reduced) ratios, but about the actual number of teeth.

Also, do you have a specific set of gears available to choose from?

 

[GlennR]

Well, in the end I would also like to end up with the most compact solution. However, there is another limitation. Z1 has to fit around a shaft of approximately 36mm. z4 should have a hole of about 50mm (not sure about the exact size yet).

Preferably I would like the distance between all three axis to be the same, it would simplify the design drastically but the sketch is just for reference and to help explain the problem. The teeth all have the same size (z1 and z2 as well as z3 and z4 have to, otherwise it won't work). With calculating gear ratio's it is possible to do it by expressing it in number of teeth, diameters or rpm, it doesn't make a difference.

I am not sure what you mean with a specific set? I can let it get custom made so all combinations or shapes are possible but I am considering to use spur gears (if this is what you mean).

 

[Dr.Peterson]

I was assuming for a school project you wouldn't have gears custom-made.

But you haven't yet said how you would judge "which are most suitable/optimal for my design". Or at least, I have no idea how the sizes you mention relate to the number of teeth.

 

[GlennR]

In the first case, for the prototype they can offcourse be 3D-printed. In the end it depends on how good our design is wether it will be improved and made from metal.

The most suitable one's fit into the dimensions I mentioned. that the gears should fit over a 36mm shaft and the other one over a 50mm shaft. Also, it should fit within a housing with a diameter of 160mm. diameter of a gear depends on the number of teeth you choose and the module.

For example:
The smallest gear should have a minimum number of teeth of 17. with a module of 1,5 (size of one tooth) the diameter of that gear becomes
d=z x m = 17 x 1,5 = 25,5mm. As you can see in the picture this is however not the diameter that has to be bigger than 36 or 50 mm for two of the four gears.


So, to conclude: with a ratio of 1/6 the gears would have a minimum number of 17 and 102 teeth.
The distance of the two axis of the gears (a) would then also be the sum of the two different diameter divided by 2.

I think this is the answer to your question. It is more a little bit of background information to understand the topic but my original question was not intented to give me an exact answer, however, the number of ratio's would maybe indeed otherwise be limitless.