Generating perfect gear ratios with number sequences.

[msd1107]

Generating perfect gear ratios with number sequences.

There has been discussion on how to select gearbox ratios. Usually you get bogged down in figuring out first gear ratio for the cars weight and hp, max ratio drop between gears, speed required in top gear, etc., etc.

But as a skeptical mathematician would say, ?Why use imprecise methods when pure number sequences would provide a result?

So let?s see if we can conjure up an elementary formula.

Assume a gearbox with ?n? ratios. Assume a factor ?m? that determines how closely the ratios are spaced. Then for a gearbox with a direct top gear, generate the ratios as follows:

n+m/n+m, n+m/n-1+m, n+m/n-2+m, ? n+m/1+m.

This looks like gibberish. Let?s generate a real set of ratios for a four speed gearbox with m=1. We get:

4+1/4+1, 4+1/4-1+1, 4+1/4-2+1, and 4+1/1+1.

This gives in real terms:

5/5 or 1.000
5/4 or 1.250
5/3 or 1.667
5/2 or 2.500

Our favorite Elan four speed transmission has ratios of 1.000, 1.231, 1.636, and 2.510. This is very close to what a theoretical number sequenced gearbox design calls for! Have we stumbled onto something? Did the Lotus gearbox designers have a numerology background?

Also, for those who have my spreadsheet, putting these ratios in gives a Figure of Merit of 100, the maximum possible. And the speed drops between gears are the same. The percentage drop between gears is 50%, 33.3%, and 25%. So the ad hoc gear ratio designs are quite similar to that produced by this numerical sequence generator, a welcome coincidence.

Let?s see what a five-speed gearbox would look like.

5+1/5+1, 5+1/5-1+1, 5+1/5-2+1, 5+1/5-3+1, 5+1/1+1

This gives in real terms:

6/6 or 1.000
6/5 or 1.200
6/4 or 1.500
6/3 or 2.000
6/2 or 3.000

But what if we want an overdrive gearbox? Let?s modify the formula as follows:

n-1+m/n+m, n-1+m/n-1+m, n-1+m/n-2+m, ? n-1+m/1+m.

So for n=5, a five speed gearbox, we get:

5-1+1/5+1, 5-1+1/5-1+1, 5-1+1/5-2+1, 5-1+1/5-3+1, 5-1+1/1+1

This gives in real terms

0.833, 1.000, 1.250, 1.667, 2.500

Again, putting these ratios in the spreadsheet gives a Figure of Merit of 100, constant speed drops between gears, and ratio drops of 50%, 33.3%, 25%, and 20%.

Manufacturers are generating gearboxes with 6, 7, and 8 ratios now, some times with 1, 2, or even 3 over drive ratios. We can generalize the formula to deal with any number of over drive ratios as follows. Define the number of over drive ratios as ?l?. then the formula becomes:

n-l+m/n+m, n-l+m/n-1+m, n-l+m/n-2+m, ? n-l+m/1+m

For a 7-speed gearbox with 2 over drive ratios we would get:

7-2+1/7+1, 7-2+1/7-1+1, 7-2+1/7-2+1, 7-2+1/7-3+1, 7-2+1/7-4+1, 7-2+1/7-5+1, and 7-2+1/7-6+1 or

6/8, 6/7, 6/6, 6/5. 6/4, 6/3, 6/2 or

0.750
0.857
1.000
1.200
1.500
2.000
3.000

Now, we can use the factor m to generate gearboxes with closer ratios. Let?s set m=2 and generate a 4-speed gearbox.

4-0+2/4+2, 4-0+2/4-1+2, 4-0+2/4-2+2, 4-0+2/1+2 or

6/6, 6/5, 6/4, and 6/3

Which is

1.000
1.200
1.500
2.000

This looks like the ultra close ratios used in racing. Again, the figure of merit is 100, there are constant speed drops between gears, and the percentage drops between gears is 20%, 25%, and 33.3%.

Well, enough for now. My spreadsheet allows for any ratio drop from 1st to 2nd instead of 50%, 33.3% etc, and to specify a speed difference increment between gears. I?ll leave it up to the more mathematically inclined to generalize the formula to incorporate more generality.

And, as usual, for the curious, request my spreadsheet so you can play around on both sides.

David
1968 36/7988
[email protected]

[tower of strength]

more mind numbing numbers David, fascinating as always and far too inteligent for my simple brain, however, tell me the number of teeth required to make the "perfect ratio set" and I'll get a price and feasability study done (theres a gear cutting firm near to where I live, and my "tame turner" buddy has a mate working there I'd like to use the MT75 as a base, because it physically fits the chassis with minimum effort and has a loverly change action with my "low budget" shifter!!

I guess we'll need to know the number of teeth in the pairs of gears that make up the train in each "gear"? (excuse my clumsy language here) so i think my next step is to pull apart a few gearboxes!!

Regards

Mark