Library / Driveline and transmission / DRV-09
Driveline and transmission
Geometric step between gears
Constant step between neighbouring gears when the ratio span is divided into equal geometric steps.
- Ref
- DRV-09
- Rev
- v1
- Field
- Driveline and transmission
- Result
- K
If a ratio span S is divided into N − 1 equal geometric steps, each step is K = S1/(N−1). Every upshift then divides the engine speed by the same factor K, so the engine returns to the same part of its power band after each shift. The ratio of each gear is ij = i1/Kj−1. Real ratio sets need not be geometric: wider steps are often used at the top to save fuel and tighter ones at the bottom.
Constant step between neighbouring gears when the ratio span is divided into equal geometric steps. Each upshift then drops the engine speed by the same factor.
Calculator
- Engine speed after an upshift from 2200 rpm1,471 rpm
- Percentage fall in engine speed per upshift33.13 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A span of 5 shared over 5 gears gives a step of 1.495 between neighbours. After each upshift from 2200 rpm the engine falls to 1,471 rpm, a drop of 33.1 %.
- Adding one gear lowers the step to 1.38, and removing one raises it to 1.71. Steps below about 1.3 give diminishing returns for the cost of an extra gear.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| N Number of gears | 2 to 12 | 1.158 | 5 | 3.842 | +1 | −7.73 % |
| S Ratio span | 1.1 to 15 | 1.024 | 1.968 | 0.9439 | +10% | +2.41 % |
Sweep of Number of gears
| Number of gears | Step between gears |
|---|---|
| 2 | 5 |
| 3.25 | 2.045 |
| 4.5 | 1.584 |
| 5.75 | 1.403 |
| 7 | 1.308 |
| 8.25 | 1.249 |
| 9.5 | 1.208 |
| 10.75 | 1.179 |
| 12 | 1.158 |
Sweep one variable
Variables and how each one contributes
Step between gears
What the equation gives you. Everything else feeds it.
Ratio span
The ratio span, first-gear ratio divided by top-gear ratio. Must be above 1. Raises the step: a wider span with the same number of gears means bigger steps. Raising it by 10 % (to 5.5) moves the result by +2.41 %, from 1.495 to 1.531.
Number of gears
The number of gears, a whole number from 2 to 12. Lowers the step: more gears share the same span in smaller steps. Raising it by 1 (to 6) moves the result by −7.73 %, from 1.495 to 1.38.
Worked example with the values above
- Exponent: 1/(N − 1) = 1/4 = 0.25.
- Raise the span: 5 to the power 0.25.
- K = 1.495.
Assumptions and limits
- A geometric ratio set with equal steps and at least two gears.
- A span of at least 1.
Common mistakes
- Using N instead of N − 1 in the exponent, which gives the step for one more gear than intended.
- Applying it where the real ratio set deliberately departs from a geometric progression.
Statistics and uncertainty
Real gearboxes have integer tooth counts, so the actual steps differ from K by one or two per cent. That is smaller than the variation caused by tyre radius and engine speed, so the geometric step is a good design target.
Where it is used
- Spacing the gears so each upshift keeps the engine in its power band.
- Estimating the step from a published ratio span and number of gears.
Maths you need
- Geometric progression: a constant ratio between terms
- Powers and roots: S raised to a fraction