If you're building an orrery, I'm guessing that you are talking about a sun gear with a planetary gear revolving about it. Is your ratio the spin of the planet gear to its revolution about the sun gear? If so, then the ratio is the same as the ratio of numbers of gear teeth of the two gears. So, for example, if the sun gear has 100 teeth and the planet gear has 25 teeth, then the ratio is 100:25 or 4:1. So the circumference of the sun gear is four times that of the planet gear and the outer gear rotates one full revolution when the connecting link revolves a quarter of the way around the sun gear.
To calculate tooth-ratios in reverse, knowing the desired motion ratio, use the "method of conjugate fractions" (fractions are the same as integer gear ratios). I was going to simply post a link to the thread where I had explained this method, but it seems to be gone, so I'll explain it again.
Take your ratio and make it a decimal between 0 and 1. If it's greater than one, then you can simply take the reciprocal (1/x). In your case this is
1/8.32 = .12019
Then we start with the conjugate fractions 0/1 and 1/1, which we know the ratio falls between. You can find the simplest fraction between two conjugate fractions by adding the numerators and denominators to create the new conjugate fraction. In our case this would be
0+1 / 1+1 = 1/2 = .500
Which is the simplest fraction between 0 and 1 and is conjugate to both the other fractions. Our ratio falls between 0/1 and 1/2, so we add the numerators and denominators of these
0+1 / 1+2 = 1/3 = .333
Now we're between 0/1 and 1/3, so
0+1 / 1+3 = 1/4 = .250
To continue
0+1 / 1+4 = 1/5 = .200
0+1 / 1+5 = 1/6 = .167
0+1 / 1+6 = 1/7 = .143
0+1 / 1+7 = 1/8 = .125
0+1 / 1+8 = 1/9 = .111
But now our ratio is greater than 1/9, so it falls between 1/9 and the next conjugate up 1/2, so we get
1+1 / 9+2 = 2/11 = .182
now our ratio is between 1/9 and 2/11, so
1+2 / 9+11 = 3/20 = .150
and so on. With each iteration you get closer and closer to your ratio and are guaranteed that you didn't miss any simpler fractions (and thus have the fewest number of gear teeth). This can be done with any ratio and to any accuracy desired.
There are also tables of gear ratios (in the Machinery's Handbook, for example) and tables of gear ratio logarithms which are also handy. But the conjugate method is just as effective.
Of course, one obvious choice, if your ratio is exactly 8.32:1, is 832:100, which reduces to 208:25. But that's a lot of gear teeth.
Hope this helps.
Don
Kansas City
