Gear Train calculation theory and example

Let's take a problem from the following text book...

Mechanics of Machinery
Dr. Mahmoud A. Mostafa
(c) 2013 by Taylor & Francis Group, LLC

Page 300, Problem 6.28

The ratio sqrt(3) [sic] is required. Use the continued fractions to obtain the number of teeth in a train such that no gear exceeds 100 teeth. The error should not exceed one part in 15 million.

‚ÄčThe above quote contains an error. The correct square root of 3 is 1.732050807568877

We want to find the gear sizes required to produce an overall speed ratio of R. Let's assume we need a three stage gear train for this.

In non-gear speak we are looking for three fractions whose product is R. Or as close to R as requested. As well, all numerators and denominators must not be greater than 100. There was no minimum number specified in the problem but a practical gear minimum is 12 teeth.

(b/a)*(d/c)*(f/e) = R ESCAPE='HTML'

There is one equation and six (6) unknowns: a, b, c, d, e, f. These represent the number of teeth on six gears. Each of these integers is limited to a range of 12 to 100.

If one proceeds with an impulsive brute-force method, one is faced with a huge number of permutations. Each of the six gears can have 89 sizes (100 - 12 + 1)

89 to the exponent 6 is approx. 497 billion ESCAPE='HTML'

Nearly four hundred ninety-seven billion! Trying out all permutations to see which one comes closest to the desired result would take a computer from a few hours to a few days of computing time.

Fortunately the Gear Train Calculator performs the calculation in much less time. It does not try out all permutations in the manner suggested. Instead it attempts to factor only candidate rationals B / A which are close to the desired ratio.

B / A = (b x d x f) / (a x c x e) ESCAPE='HTML'

This may seem like a trivial distinction from the previous brute-force approach. However, we now are considering only a single fraction B / A whose numerator and denominator need to be factorable into three factors each. The number of permutations that need to be considered are now dramatically reduced:

 ESCAPE='HTML'

Therefore A can range from 1,728 to 577,350. This means we will test 575,623 fractions. A lot less than 497 billion.

The numerator B is always A x 1.732050807568877, rounded to the nearest integer. We divide out that fraction to check how close it is to that ratio. If the result is fairly close (for example, closer than a previous estimate) then we try to factor B and A into three factors each that satisfy our gear limits.

For the final result the calculator re-factors the best found fraction B / A again to find the optimum gears. All possible factorizations are evaluated. It is desirable to get the lowest total tooth count. Therefore the sum of b + d + f needs to be minimized as well as the sum a + c + e.

Now let's put theory into practice. Enter in the ratio. Formula sqrt(3) is accepted. Set Stages to 3. Set the gear size limits. Press "Calculate". You will see the result below.

Gear Calculator screen shot ESCAPE='HTML'

The result can be interpreted as:

(53/73)x(51/37)x(45/26) = 1.732050807393273 ESCAPE='HTML'

This result is more accurate than the book problem demanded. Perhaps more accurate than the continued-fractions method would have been able to produce. The Calculator uses an exhaustive search that is simply not feasible to do by hand but it is guaranteed to give the best result possible.

Revised 8 Apr 2015