Driving resistances:
While driving, the car is subject to various resistances:
- Rolling resistance
- Gradient resistance
- Air resistance
- Acceleration resistance
These resistances must be overcome in order to maintain speed. The force required to do this is called Frij; this is the sum of all driving resistances.
The rolling resistance increases at higher speeds due to deformation of the tyres, the gradient resistance only applies when there is an incline (on a flat road this is therefore 0), and the air resistance is very low at low speeds. As the driving speed increases, the air resistance increases quadratically. Air resistance plays the biggest role when we look at the total driving resistances.
On this page, the individual driving resistances are calculated up to the total driving resistance (Frij).
Rolling resistance:
Rolling resistance is caused by various factors such as tyre deformation, wheel alignment (and with it the lateral movement of the tyre) and the type of road surface. The degree to which the tyres deform depends on the type of tyre. The more “smoothly” the tyre can roll over the road surface (so experiencing as little resistance as possible), the less force is needed to keep the wheel moving and the lower the fuel consumption will be. On a deformable (soft) surface, such as sand or mud, rolling resistance increases further due to additional frictional forces between the tyre and the surface, and due to the permanent deformation of the surface itself.
The table alongside shows that the rolling resistance coefficient is low (0.010) on dry asphalt and high (up to 0.3) on sand. The table is based on a low to medium vehicle speed (up to 80 km/h) at which the rolling resistance of different types of tyres is fairly constant.
At these speeds, the influence of speed on rolling resistance is negligible.
Factors that influence rolling resistance:
- Slip (during braking or accelerating): in the contact patch, slip occurs when the tyre transmits forces. With light slip (such as during gentle acceleration) the rolling resistance can even temporarily decrease. With strong slip (during hard acceleration or heavy braking) the rolling resistance increases, because the deformation behaviour changes and more energy loss occurs.
- Wheel alignment (toe and camber): incorrect alignment causes extra lateral forces (for example with toe-in or camber). These forces increase rolling resistance. This is because the wheel no longer rolls perfectly straight ahead, but builds up a lateral load. The influence becomes greater with larger alignment deviations.
- Temperature: When the tyre starts rolling, it heats up. As a result, the material properties of the rubber change: damping decreases, stiffness changes, and the tyre deforms less. This causes rolling resistance to decrease as the temperature rises. After a few minutes the temperature reaches equilibrium, where heat generation and heat dissipation are balanced.
- Speed: At higher speed more deformations per second occur (more revolutions), which leads to higher temperature and greater deformation forces. As a result, rolling resistance increases, especially with tyre types that have higher hysteresis such as standard or winter tyres.
At low speeds (up to approx. 80 km/h) the rolling resistance coefficient remains largely constant. The table with values per surface shown at the top of this paragraph is based on these speeds. At higher speeds, rolling resistance increases, especially with standard and winter tyres. Mathematically, the increase in rolling resistance can be approximated with a quadratic relationship. The graph below shows this effect:
- SR tyres (standard tyres) have relatively high hysteresis and show the strongest increase in rolling resistance at higher speeds.
- M+S tyres (winter tyres) rank between SR and HR tyres in terms of performance. These tyres have extra tread and sipes, which increases rolling resistance.
- HR tyres (high performance) have reinforced carcasses and rubber compounds with low hysteresis. They are the most efficient at high speed and show the smallest increase in rolling resistance.”
When the rolling resistance coefficient and the vehicle weight are known, the rolling resistance can be calculated. The following data are known:
- BMW X3 with a mass (m) of 1700 kg;
- Gravitational acceleration (g) is: 9.81 m/s^2;
- Friction coefficient (μ) is: 0.010;
- Horizontal road surface.
First we multiply the vehicle mass by the gravitational acceleration to calculate the normal force (Fn):
We then multiply the normal force by the rolling resistance coefficient to calculate the force required to overcome the rolling resistance of the tyres on the road surface.
Gradient resistance:
When a vehicle drives up an incline, a so‑called gradient resistance occurs. This resistance arises because part of the gravitational force acts against the direction of travel. Extra force from the engine is therefore needed to drive the vehicle uphill at constant speed or with acceleration.
When driving uphill, the full gravitational force is no longer exerted perpendicular to the road surface, but partly along the incline. This changes the distribution of the forces on the vehicle:
- The part perpendicular to the road surface determines the normal force (Fn), which influences rolling resistance.
- The part parallel to the road surface provides the gradient resistance (Fhelling).
Over a distance of 100 metres, the vehicle has climbed 5 metres (see image). That means the gradient is 5%. We calculate the gradient angle using the tangent (tan).
Calculating tan α:
tan α = opposite / adjacent = 5 / 100
α = tan⁻¹(5/100) = 2.86°
Tip: on the calculator, press shift and then the tan button to get tan ̄ ¹, and put (5/100) in brackets. The result can be displayed in degrees or radians, depending on your calculator settings. To convert from radians to degrees, use the following formula:
Degrees = Radians * (180 / π)
Rolling resistance becomes slightly lower when driving up an incline, because the normal force decreases. A smaller part of the gravitational force then acts perpendicular to the road surface, so the tyres press less hard on the road. This leads to less deformation and therefore less rolling resistance.
In the formula for rolling resistance this is represented as follows (the cosine of the gradient angle determines how much force still acts perpendicular):
The effect on rolling resistance is small, for example only 0.21 N in this example, and is neglected in most practical situations. We can calculate the gradient force (Fhelling) by multiplying the normal force (Fn) by the gradient angle. We call the angle sine (sin) alpha. The sine of the gradient angle determines how much of the gravitational force acts along the incline.
A force of just over 832 Newton + the rolling resistance of 166.56 N is required to drive up the incline. We can also combine the formulas for rolling and gradient resistance, because the gradient also influences the rolling resistance. Note that air resistance has not yet been included here, so this is not yet the total driving resistance! This will be covered further down this page.
Air resistance:
While driving, the vehicle experiences resistance from the air flow. This is called air resistance. As speed increases, air resistance increases quadratically. As a result, the vehicle will accelerate less and less as vehicle speed increases.
When driving on a provincial road, the difference in fuel consumption between 60 and 80 km/h will be minimal. The difference in consumption between 120 and 140 km/h is much greater due to the increasing air resistance. Fuel consumption is often most favourable around 90 km/h as a result of the ideal engine speed range in the highest gear; see the page about the specific fuel consumption.
The force required to overcome air resistance can be calculated as follows:
Explanation of the formula:
½ = one half, which can be entered in the calculator as 0.5;
ρ = Rho. This indicates the density. In this case the density of air, with the unit m³;
Cw = drag coefficient. For a passenger car, the Cw value is between 0.25 and 0.35. For a truck between 0.65 and 0.75;
A = frontal area of the car (determined in the wind tunnel) in m²;
v² = the speed of the vehicle squared, with the unit m/s;
For this calculation we use the following data:
- ρ = 1.28 kg/m³ (depending on temperature and humidity)
- Cw = 0.35
- A = 1.8 m²
- v² = 100 km/h = (100 / 3.6) = 27.78 m/s² (metres per second squared because this concerns an acceleration):
Using the known data we fill in the formula for Flucht:
A force of 311.11 N is therefore required to overcome air resistance.
Acceleration resistance:
During acceleration or deceleration, acceleration resistance is created. Force in Newton is required to overcome this acceleration resistance. We again take the BMW X3 with a vehicle mass of 1700 kg as our example.
The required force to overcome the acceleration resistance (Facceleration) depends on the vehicle mass (m) and the acceleration or deceleration (a) in Newton. In this example, we assume a minimum deceleration of
0.2 m/s^2. The required force to overcome the acceleration resistance can be calculated using the following formula:
Total driving resistance:
The total driving resistance (Fdrive) is the sum of all previously mentioned resistances. The rolling resistance + the gradient resistance + the air resistance together make up Fdrive:
Conclusion: to drive up a 5% gradient at 100 km/h with an acceleration rate of 0.2 m/s² in windless conditions (0 BFT), the total required force amounts to 1,649.78 Newton.
Not only the driving resistances, but also the efficiencies and the reductions in the gearbox are important for the designer to calculate in advance.
The gearbox and the gear ratios are matched to the characteristics of the engine. This is described on the page overbrengingsverhoudingen.
