Library / Driveline and transmission / DRV-04
Driveline and transmission
Minimum first-gear ratio for climbing
Smallest overall ratio in first gear that lets the engine’s maximum torque produce the force needed to climb the steepest slope at low speed.
- Ref
- DRV-04
- Rev
- v1
- Field
- Driveline and transmission
- Result
- i1
At low speed the engine’s maximum torque Tmax, multiplied by the ratio and the efficiency, gives the wheel torque, and dividing by the rolling radius r gives the force. Setting that equal to the climbing force Fclimb gives the smallest overall ratio i1 that can do the job. For steady slow climbing Fclimb = m g (f cos θ + sin θ). A larger ratio does no harm to the climb, and excess ratio does not force wheelspin if the engine torque is reduced. A traction-limited ratio bound applies only at the specified applied torque.
Smallest overall ratio in first gear that lets the engine’s maximum torque produce the force needed to climb the steepest slope at low speed. A lower ratio cannot reach the required wheel force.
Calculator
- Wheel torque needed, F r19.25 kN m
- Engine torque needed at an overall ratio of 12 (below the minimum)1,782 N m
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.
- Climbing with 35 kN at a rolling radius of 0.55 m needs a first-gear overall ratio of at least 14.26, given 1,500 N m of engine torque and 0.9 efficiency. A lower ratio such as 12 would need 1,782 N m, more than the engine can give.
- The ratio found is a lower bound for climbing. The upper bound comes from traction: wheel force cannot exceed the friction limit, so the bound only holds at the engine torque you actually apply.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Tmax Maximum engine torque | 200 to 4,000 N m | 5.347 | 106.9 | 101.6 | +10% | −9.09 % |
| Fclimb Climbing force | 5 to 100 kN | 2.037 | 40.74 | 38.7 | +10% | +10 % |
| r Rolling radius | 0.25 to 0.9 m | 6.481 | 23.33 | 16.85 | +10% | +10 % |
| η Driveline efficiency | 0.5 to 1 | 12.83 | 25.67 | 12.83 | +10% | −9.09 % |
Sweep of Climbing force
| Climbing force (kN) | Minimum first-gear ratio |
|---|---|
| 5 | 2.037 |
| 16.88 | 6.875 |
| 28.75 | 11.71 |
| 40.63 | 16.55 |
| 52.5 | 21.39 |
| 64.38 | 26.23 |
| 76.25 | 31.06 |
| 88.13 | 35.9 |
| 100 | 40.74 |
Sweep one variable
Variables and how each one contributes
Minimum first-gear ratio
What the equation gives you. Everything else feeds it.
Climbing force
The force needed to climb the steepest required slope at low speed, in kN. Raises the minimum ratio in proportion. Raising it by 10 % (to 38.5) moves the result by +10 %, from 14.26 to 15.69.
Rolling radius
The loaded rolling radius of the tyre, in m. Raises the minimum ratio in proportion, because a larger wheel gives less force for the same torque. Raising it by 10 % (to 0.605) moves the result by +10 %, from 14.26 to 15.69.
Maximum engine torque
The engine’s maximum torque, in N m. Lowers the minimum ratio in inverse proportion: a stronger engine needs less gearing. Raising it by 10 % (to 1,650) moves the result by −9.09 %, from 14.26 to 12.96.
Driveline efficiency
The driveline efficiency from engine to ground, between 0 and 1. Lowers the minimum ratio a little: a more efficient driveline needs less gearing. Raising it by 0.05 (to 0.95) moves the result by −5.26 %, from 14.26 to 13.51.
Worked example with the values above
- Wheel torque needed: F r = 35 kN × 0.55 m = 19.25 kN m.
- Divide by the engine torque and efficiency: 19,250 / (1,500 × 0.9).
- i1 ≥ 14.26.
Assumptions and limits
- Steady low-speed climbing, so air drag and inertia are negligible.
- The torque is the engine’s maximum, available at the engine speed that corresponds to the vehicle speed.
Common mistakes
- Using the unloaded wheel radius rather than the rolling radius.
- Choosing the ratio from the climb alone without checking that the ground can supply the force.
Statistics and uncertainty
The climbing force depends on the rolling coefficient and slope, each uncertain by 10 to 20 %, so the minimum ratio inherits that. Choosing the ratio about 10 % above the minimum gives a margin for losses, altitude derating and soft ground.
Where it is used
- Choosing the first-gear and final-drive ratios for a gradient requirement.
- Checking an existing gearbox against a new slope requirement.
Maths you need
- Torque and force: force equals torque over radius
- Overall ratio: gearbox times final drive