Library / Driveline and transmission / DRV-02
Driveline and transmission
Total tractive force required
Force the driven wheels must deliver to overcome rolling resistance, the slope, air drag and the inertia of the vehicle and its spinning parts.
- Ref
- DRV-02
- Rev
- v1
- Field
- Driveline and transmission
- Result
- Freq [N]
The driven wheels must supply force to overcome four things: rolling resistance f m g cos θ, the slope component m g sin θ, air drag ½ρCDA v², and the inertia of the vehicle and its spinning parts, m(1 + λ)a, where λ is the equivalent rotating-mass fraction. The drag expression is a magnitude opposing the motion, with v the speed relative to the air. Grade percentage is 100 tan θ. When the vehicle is held at constant speed, a is zero. Drag area CDA is the drag coefficient times the frontal area.
Force the driven wheels must deliver to overcome rolling resistance, the slope, air drag and the inertia of the vehicle and its spinning parts. It is the load line that every gear and engine choice is checked against.
Calculator
- Rolling resistance, f m g cos θ2,932 N
- Slope force, m g sin θ8,550 N
- Air drag, ½ρCDA v²675 N
- Inertial force, m(1 + λ)a5,500 N
- Grade, 100 tan θ8.749 %
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 10,000 kg vehicle on a 5° slope at 15 m/s with 0.5 m/s² acceleration needs 17.66 kN. The slope takes 48.4 % of that, inertia 31.1 %, rolling 16.6 % and air drag only 3.8 %.
- Doubling the air speed to 30 m/s would raise the drag from 675 to 2,700 N, because it goes with the square, which is why drag dominates at high speed and the slope dominates at low speed.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| θ Slope angle | −30 to 35 ° | −40,330 | 64,850 | 105,200 | +1.3 ° | +12.5 % |
| m Vehicle mass | 1,000 to 60,000 kg | 2,373 | 102,600 | 100,200 | +10% | +9.62 % |
| a Acceleration | −3 to 3 m/s² | −20,840 | 45,160 | 66,000 | +0.12 m/s² | +7.48 % |
| f Rolling resistance coefficient | 0.01 to 0.2 | 15,700 | 34,270 | 18,570 | +10% | +1.66 % |
| v Relative air speed | 0 to 40 m/s | 16,980 | 21,780 | 4,800 | +10% | +0.803 % |
| λ Rotating-mass fraction | 0 to 0.5 | 17,160 | 19,660 | 2,500 | +10% | +0.283 % |
| CDA Drag area | 0.5 to 10 m² | 17,050 | 18,330 | 1,283 | +10% | +0.382 % |
| ρ Air density | 0.9 to 1.3 kg/m³ | 17,490 | 17,710 | 225 | +10% | +0.382 % |
Sweep of Slope angle
| Slope angle (°) | Tractive force required (N) |
|---|---|
| −30 | −40,330 |
| −21.88 | −27,640 |
| −13.75 | −14,280 |
| −5.625 | −511.7 |
| 2.5 | 13,390 |
| 10.63 | 27,160 |
| 18.75 | 40,500 |
| 26.88 | 53,150 |
| 35 | 64,850 |
Sweep one variable
Variables and how each one contributes
Tractive force required
What the equation gives you. Everything else feeds it.
Rolling resistance coefficient
The rolling resistance coefficient, a plain number: about 0.01 to 0.02 on a hard road and up to 0.2 on soft ground. Raises the rolling part of the force in proportion; the total moves less because the slope and inertia parts do not change. Raising it by 10 % (to 0.033) moves the result by +1.66 %, from 17,660 to 17,950 N.
Vehicle mass
The vehicle mass, in kg. Raises the rolling, slope and inertia parts in proportion to the mass; the air drag does not change. Raising it by 10 % (to 11,000) moves the result by +9.62 %, from 17,660 to 19,350 N.
Slope angle
The slope angle in degrees, positive uphill. A 30 % grade is about 16.7°. Raises the slope part, m g sin θ, which is the largest term at the default. Raising it by 2° (to 7°) moves the result by +19.2 %, from 17,660 to 21,050 N.
Air density
The air density in kg/m³: about 1.2 at sea level and falling with altitude and heat. Raises the air drag in proportion, which is a small part of the total at the default speed. Raising it by 10 % (to 1.32) moves the result by +0.382 %, from 17,660 to 17,720 N.
Drag area
The drag area, drag coefficient times frontal area, in m². A box-shaped military truck is around 4 to 8 m². Raises the air drag in proportion. Raising it by 10 % (to 5.5) moves the result by +0.382 %, from 17,660 to 17,720 N.
Relative air speed
The speed relative to the air, in m/s. With no wind it equals the vehicle speed. Raises the air drag with the square of the speed. Raising it by 10 % (to 16.5) moves the result by +0.803 %, from 17,660 to 17,800 N.
Rotating-mass fraction
The equivalent rotating-mass fraction: the extra inertia of the wheels, shafts and engine, as a fraction of the vehicle mass. It is larger in low gears, about 0.05 to 0.3. Raises only the inertial part, which acts only while accelerating. Raising it by 0.05 (to 0.15) moves the result by +1.42 %, from 17,660 to 17,910 N.
Acceleration
The acceleration in m/s², positive when speeding up. Use a negative value for deceleration. Raises the inertial part in proportion to the acceleration; at the default it is the third-largest term. Raising it by 0.5 m/s² (to 1 m/s²) moves the result by +31.1 %, from 17,660 to 23,160 N.
Worked example with the values above
- Rolling and slope: 2,932 + 8,550 = 11,480 N.
- Air drag and inertia: 675 + 5,500 = 6,175 N.
- Freq = 17,660 N (17.66 kN).
Assumptions and limits
- Steady slope and wind, with the drag area and rolling coefficient constant over the speed range.
- Standard gravity 9.81 m/s²; the force is the total at the contact patches, ahead of any traction limit.
Common mistakes
- Using the slope angle in the rolling term as if it were a percentage grade.
- Applying the inertial term when the vehicle is cruising, where a is zero.
Statistics and uncertainty
Rolling resistance and the rotating-mass fraction are the least well known inputs; the rolling coefficient can vary by a factor of 2 with surface and tyre pressure. A 30 % error in f here changes the total force by about 5 %, so sizing needs margin.
Where it is used
- Sizing the engine, gearbox and traction for a climb, a top speed or an acceleration target.
- Seeing which resistance dominates at a given speed and slope.
Maths you need
- Forces on a slope: components along and across the slope
- Drag force: ½ρCDA v²