A while ago, trying to build some kind of an orrery, I've had a problem of finding a combination of Technic gears that yields a very specific and precise gearing ratio. To help myself out, I've written a simple program, and hope it may help other builders facing a similar problem too.
So, basically, what it does is ask for the desired total gearing ratio, maximum number of gears involved, and the information which gears (not) to use, then display the gear combination that satisfies the input parameters, or at least comes closest possible under given limits. I have to admit it is a bit crude on the surface, but it works, is simple to use, and does not mutilate your computer with registry bloat. The full instructions and notes are packed in a Readme.txt file in the ZIP available for download, but let's just run quickly through them here.
There is no particular installation; just unpack the downloaded LegoGearCalc.zip in a folder of your choice, and start LegoGearCalc.exe. A text window should open, asking you for a first parameter: the desired gear ratio, that is, the number of turns you want the output axle to do while the input axle does exactly one full turn. This can be an integer, a decimal number, or even an expression using basic operators. E.g. entries such as 14, 0.18, 6/13, (5+2)/3*9 etc. will work fine.
Then, enter the number of gear pairs you would like to involve at most. Default is four, but you can search up to five pairs. Finally, specify any of the gears you would like to exclude from the search by entering their numbers of teeth separated by commas. Worm gear is 1. Just hit Enter (leave the entry empty) to use all Technic gears.
After a few moments at most, the program will reply with a list of gear pairs that come closest to the desired ratio, or an exact solution if there is one. Occasionally there are several ways to reach the same result, or it can be reached using less gear pairs than set as a maximum. Of course, in your system they need not be in the order from the list. Two screenshots of typical usage examples are below, with the entries typed by the user in cyan.
Many gear combinations are not easy to mesh together since they require specific spacing, but there are various methods of creating a non-integer stud distances between axle holes one can find (or better yet, invent).
(Or click here to view or download just its instruction text file.)