Library / Driveline and transmission / DRV-10
Driveline and transmission
Ratio of gear j in a geometric set
Overall ratio of the jth gear when every step is the same factor K.
- Ref
- DRV-10
- Rev
- v1
- Field
- Driveline and transmission
- Result
- ij
With a constant step K between neighbouring gears, the first-gear ratio i1 is divided by K once for each gear above first, so ij = i1/Kj−1. The engine speed just after an upshift is the speed before it divided by K. Gear 1 is the largest ratio. Real ratio sets need not be exactly geometric.
Overall ratio of the jth gear when every step is the same factor K. Gear 1 is the largest ratio, and each higher gear divides it by K once more.
Calculator
- Ratio of the next gear up, ij+14.148
- Engine speed after an upshift from 2000 rpm1,333 rpm
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.
- With a first gear of 14 and a step of 1.5, gears 1 to 5 have ratios 14, 9.33, 6.22, 4.15, 2.77. Gear 3 has 6.222.
- Each upshift divides the engine speed by 1.5: from 2000 rpm it falls to 1,333 rpm. A tighter step gives a smaller fall, so the engine stays near its power peak.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| i1 First-gear ratio | 3 to 40 | 1.333 | 17.78 | 16.44 | +10% | +10 % |
| j Gear number | 1 to 10 | 0.3642 | 14 | 13.64 | +1 | −33.3 % |
| K Step between gears | 1.1 to 2 | 3.5 | 11.57 | 8.07 | +10% | −17.4 % |
Sweep of Gear number
| Gear number | Ratio of gear j |
|---|---|
| 1 | 14 |
| 2.125 | 8.872 |
| 3.25 | 5.622 |
| 4.375 | 3.563 |
| 5.5 | 2.258 |
| 6.625 | 1.431 |
| 7.75 | 0.9068 |
| 8.875 | 0.5747 |
| 10 | 0.3642 |
Sweep one variable
Variables and how each one contributes
Ratio of gear j
What the equation gives you. Everything else feeds it.
First-gear ratio
The overall ratio of first gear, including the final drive. Raises every gear’s ratio in proportion. Raising it by 10 % (to 15.4) moves the result by +10 %, from 6.222 to 6.844.
Step between gears
The geometric step between neighbouring gears. Must be above 1; typical values are 1.3 to 1.8. Lowers the ratio of gear j when raised, because the ratio is divided by K j−1 times. Raising it by 0.1 (to 1.6) moves the result by −12.1 %, from 6.222 to 5.469.
Gear number
The gear number, a whole number from 1 (first) upwards. Lowers the ratio when raised: each higher gear divides it by K once more. Raising it by 1 (to 4) moves the result by −33.3 %, from 6.222 to 4.148.
Worked example with the values above
- Divide by K once for each gear above first: Kj−1 = 1.52 = 2.25.
- Divide: 14 / 2.25.
- ij = 6.222.
Assumptions and limits
- A geometric set: the same step K between every pair of neighbouring gears.
- Overall ratios in the engine-over-wheel convention.
Common mistakes
- Dividing by Kj instead of Kj−1, which shifts every gear by one position.
- Counting the gear number from zero.
Statistics and uncertainty
Tooth counts are whole numbers, so a real gear’s ratio differs from the ideal geometric value by up to a few per cent. The effect on speed is small compared with changes in tyre radius, and it is usually accepted.
Where it is used
- Writing out the full ratio table of a gearbox from the first gear and the step.
- Predicting engine speed after a shift.
Maths you need
- Geometric progression: a constant ratio between terms
- Powers: repeated multiplication