Library / Driveline and transmission / DRV-08
Driveline and transmission
Number of gears for a ratio span
Smallest number of gears that covers a given ratio span without any single step between neighbouring gears exceeding the allowed maximum.
- Ref
- DRV-08
- Rev
- v1
- Field
- Driveline and transmission
- Result
- N
To cover a span S with steps no bigger than Kmax between neighbouring gears, the number of steps is the smallest whole number n with Kmaxn ≥ S, which is ceil(ln S/ln Kmax). Adding the first gear gives N = 1 + ceil(ln S/ln Kmax). The span must be at least 1 and the maximum step above 1. If S = 1, one ratio is enough for this idealised span. Real ratio sets are not always geometric, but a geometric set is the usual starting point.
Smallest number of gears that covers a given ratio span without any single step between neighbouring gears exceeding the allowed maximum. The result is always a whole number.
Calculator
- Ideal (fractional) number of steps, ln S/ln Kmax3.424
- Actual step with this number of gears, S1/(N−1)1.495
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 with no step above 1.6 needs 5 gears: ln 5/ln 1.6 = 3.424, which rounds up to 4 steps.
- Allowing steps up to 2 gives 4 gears for the same span, and limiting steps to 1.3 needs 8. Each extra gear costs weight and shifts, so step size is a trade-off with the engine’s useful speed range.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Kmax Largest allowed step | 1.1 to 2 | 4 | 18 | 14 | +10% | −20 % |
| S Ratio span (at least 1) | 1.01 to 15 | 2 | 7 | 5 | +10% | 0 % |
Sweep of Ratio span (at least 1)
| Ratio span (at least 1) | Number of gears |
|---|---|
| 1.01 | 2 |
| 2.759 | 4 |
| 4.508 | 5 |
| 6.256 | 5 |
| 8.005 | 6 |
| 9.754 | 6 |
| 11.5 | 7 |
| 13.25 | 7 |
| 15 | 7 |
Sweep one variable
Variables and how each one contributes
Number of gears
What the equation gives you. Everything else feeds it.
Ratio span (at least 1)
The ratio span, first-gear ratio divided by top-gear ratio. Must be above 1. Raises the count only when it crosses a threshold at which another gear is needed; otherwise the answer does not change. Raising it by 20 % (to 6) moves the result by +0 %, from 5 to 5.
Largest allowed step
The largest step the design allows between neighbouring gears: about 1.3 to 1.8 for road vehicles, up to 2 for off-road gearboxes. Must be above 1. Lowers the count when raised, but only in whole-number jumps. Raising it by 0.1 (to 1.7) moves the result by +0 %, from 5 to 5.
Worked example with the values above
- Natural logarithms: ln S = 1.609 and ln Kmax = 0.47.
- Ratio, rounded up: 3.424 → 4 steps.
- N = 1 + 4 = 5 gears.
Assumptions and limits
- A geometric ratio set with equal steps.
- Span at least 1 and maximum step above 1.
Common mistakes
- Forgetting to round up, which gives a fractional number of gears.
- Counting steps instead of gears. N is the number of gears, one more than the number of steps.
Statistics and uncertainty
The count is a whole number, so it is not sensitive to small changes in S except near a threshold. If S is only known to ±15 %, check whether the count changes across that range: if it does, either choice may do and the final choice rests on cost or packaging.
Where it is used
- Deciding between a 4-speed, 6-speed or 8-speed transmission.
- Checking whether an existing gearbox covers a required span.
Maths you need
- Logarithms: ln S/ln K gives the number of steps
- Rounding up: the ceiling function