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I cannot seem to get this right.. I want to create 2 gears that work together.

enter image description here

I can create gears, but I cannot seem to figure out the math needed to make them uniform, or where they work together.

There is a mathematical way to do this. I do not know the formula..

What I'm trying to accomplish.

not my artwork

What I tried:

  • Create a circle 800px x 800px
  • Create a circle 600px x 600px
  • Create a star with inner radius 200 (400 diameter), 500 (1000 diameter) outer, with 12 points.
  • Compound the smaller circle and star. Intersect the remainder.

Do the same thing, except half each value and the points.

I would think this would make a 2:1 gear ratio where both gears match exactly, but they don't...

enter image description here

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Please note that on your picture the gears don't match... They are two different gears: you just create another brush with the part of the first one –  Ilan May 22 at 11:11
    
The match depends on the number of teeth you want to have on the gears –  Ilan May 22 at 11:14
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This question appears to be off-topic because it is about engineering. Should ask on an appropriate science/engineering board for the formula and then come back here if you struggle with creating it in Illustrator. –  Ryan May 22 at 11:37
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Perhaps this question should be closed or migrated? –  joojaa May 22 at 12:22
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I asked a related question about drawing gears with Bezier curves, and I found useful resources in the answers: math.stackexchange.com/questions/208153/… . Please note that this is from a programmers point of view initially, but you may find something useful there.. –  Antoine Lecaille May 22 at 21:30
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4 Answers 4

up vote 5 down vote accepted

The same way one would create a real working gear. Here are tools which will create gears:

See the site http://woodgears.ca/ for more information on gears --- they have a Gear template generator as well: http://woodgears.ca/gear_cutting/template.html

Alternately, you can draw them by hand --- you just need to use basic geometry to ensure that the diameter of each gear is an integral ratio.

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The gear generator works. I'll just use the trace tool to re-create it as vector. I really thought the answer would be more simple. –  kcdwayne May 22 at 14:28
    
Use one of the tools which will create a .dxf and import that into Illustrator instead. –  WillAdams May 22 at 14:29
    
I tried that first. Unfortunately, it imports the text list instead of the graphic. –  kcdwayne May 22 at 14:30
    
There's an option to d/l HPGL --- use that and then use pstoedit to convert it to whatever you want. –  WillAdams May 22 at 14:33
    
@kcdwayne The irony of it all. –  Ryan May 22 at 16:15
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I'm not a graphic designer or user of Illustrator, but I am a mechanical engineer. I won't bother going into the details of a properly engineered gear, but just how to get yours to "look right".

Your teeth do not have the same depth as each other because you are simply cutting all your dimensions in half. Instead, the bases and tips of the teeth need to be the same distance away from each other in both sizes of gear. The dimension that should be cut in half is the diameter of the "pitch circle".

The pitch circle is somewhere between the tips and bases of the gears. You can just put it halfway between for your purposes. (In properly designed gears for minimal noise and backlash, etc., there is a complicated formula to determine where to position the pitch circle and how to contour the teeth accordingly--see @joojaa's answer.)

So for the big gear where the two circles are at 600 and 800 diameter, your pitch circle is 700 in diameter with the tips and bases +/-100 in diameter. So your small gear should be 350 in diameter with the tips and bases +/-100 in diameter. Therefore, your small gear should use circles of 250 and 450 in diameter.

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UPDATE: the mathematics of the GEARS explained in this WIKI ARTICLE

Using PEN tool create the small shape as in the picture

Drag it to Brushes panel and in the popup window choose Pattern Brush + Stretch to fit

Apply to circle as on the picture

1 minute operation

Probably you will need to adjust the size, the angles, the stroke etc

enter image description here

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Those graphics don't seem to match up.. –  kcdwayne May 22 at 10:50
    
@kcdwayne I don't understand your comment. Explain your question better with screenshots and points of trouble and we will be glad to help you –  Ilan May 22 at 10:54
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@Ilan, this is not a gear that would work all that well in reality. Its typical for art tough. –  joojaa May 22 at 11:25
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A typical gear form the 1200's might have looked a bit like that but modern gear would be a circle involute or a cycloid. I'm trying to summarize this at the moment –  joojaa May 22 at 11:35
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For simple gear illustrations, this is a really good simple method as it will apply teeth of the same size on any size of circle and is easily adjustable, for example i.imgur.com/LG9krpS.png –  user568458 May 23 at 10:24
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This may be out of focus for GD.se but i will try to birefly explain this.

There are several kinds of gear functions. The most common being the involute of a circle and the cycloid. In principle you could generate a corresponding gear shape pair for nearly any shape but in practice involute has superior properties for misalignment. This is because the pair of a circle involute is a circle involute.

The most common gear shape.

Involute gears are by far most common. The involute shape is simply the sum of 2 vectors. Vector one is rotating along the base circle that the curve is the involute of. Vector 2 is perpendicular to this vector and is the length of how far has been traveled on the base circle.

enter image description here

Image 1: Formation of the involute shape.

This in mathematical parametric terms forms a function for x and y as for example as follows, you get the following parametric function:

x = r * sin(t) - r * t * cos(t)
y = r * cos(t) + r * t * sin(t)

Where t is the angle in radians and r is the radius (If you graph 0 to - Pi/2 you get the same image). Okay but this sin itself is not enough. As you need to calculate the shift and size for the mirror of this curve to get the teeth aligned. But this was a very short version of my introductory lecture on the subject. In any case pick a book on mechanical engineering for the really gritty details.

As it is now i think this over exceeds the purpose of this exchange. But before i part ways ill leave a hint on how to plot the curve in with adobe tools in form of a plotting routine for Photoshop (could be easily changed for illustrator). See this post:

Trace equation of line in Photoshop

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