gear size selection for a compound gear system to fulfill a specific output ratio

Question

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.

Answer 1

The solution that comes to mind for me would be to start with your target gear ratio and create a fractional form of the target gear ratio, in this case $\frac{781}{100}$.

At this point, that numerator is far too big, so you would factor 781 and 100 and find a combination of the factors that does the job.

When you have gears in a train, the places where the gears mesh are where we need to consider the gear ratio, so in the case of 7.81 we can consider $\frac{781}{100} = \frac{11*71}{10*10} = \frac{11}{10}*\frac{71}{10}$ which means you need to have a place that the gears mesh for each of those reductions. This could mean a driving gear with 10 teeth driving a double gear with the driven gear having 11 teeth and the attached gear having 10 teeth which is then driving a gear with 71 teeth.

Thinking of it as $\frac{drivenA}{drivingA} * \frac{drivenB}{drivingB} = G.R.$ and $\frac{double}{driver}*\frac{driven}{double}$ where A and B are your needed gear ratios and a double gear has one driving gear and one driven gear from different ratios can be a useful way to make a gear train as you chain these together.

Answer 2

This is an NP-complete problem. Therefore no great quickie algorithm exists.

The complexity class of decision problems for which answers can be checked for correctness, given a certificate, by an algorithm whose run time is polynomial in the size of the input (that is, it is NP) and no other NP problem is more than a polynomial factor harder. [Google]

Geartrain Combinations Calculator: https://scientific601.altervista.org/gear/gearcalc.html So, in the end, it isn't a matter of me learning how to solve it firsthand, it's a matter of seeing and understanding the java algorithm written to solve it. Only some optimizations are used to achieve the computations within a reasonable time-span. I've come to better solutions than the program could generate at times. A good use-case for a quantum computer if absolutes and high decimal accuracy is needed.

Code at: https://scientific601.altervista.org/gear/gearcalc.html