I need help creating a compound gear system that will result in the sought after drive ratio or very close to it. Naturally gear teeth need to be whole numbers. Websites like 'GearGenerator' (dot-com) help me solve my problem via guesswork. I am looking to solve my problem with the appropriate application of mathematics be it Algebra or Differential Equations. I'll give you an example of one of several gear ratios I am after.
Example A- Constraints: gears cannot be less than 15 teeth(m1), gear cannot exceed 100 teeth(m1).
I need to use a compound stud gear system to result in a ratio of 7.81 Presently the solution that does not suit my constraints is a 20 tooth driving gear and a 156 tooth driven gear. I know I can guess around and then make minor adjustments to come up with a four to six gear system. I do have several more oddball ratios to solve and I would like to approach this with the aim of becoming a better engineer by applying physics and mathematics.
So, how do I go about this problem?
Sample A- An example of a solution to an easier ratio of 2.55 is as follows: https://geargenerator.com/#225,312.5,50,1,1,2,8288.700000012579,4,1,20,5,4,27,0,0,0,0,0,0,0,20,5,4,27,-86,0,0,0,0,1,0,51,12.75,4,27,0.3000000000000007,0,0,0,0,2,1,20,5,4,27,5.300000000000001,0,0,0,0,0,1,3,-229 On the top left is a driving gear of 20 teeth, then another gear of the same size(functionally doing nothing but reversing direction), then the 51 tooth gear driving a gear on a shared axle. This results in a sought ratio of 2.55.
Please Note: -Having a close answer to my question will help me have a more accurate celestial model. That is why 'meh, close enough' is not an ideal answer to give myself. Also, I want to learn.
-I'd appreciate your patience and understanding with my cavalier word choice and casual engineering background. Please don't hold back on presenting me solutions no matter the complexity. I have time.