# How to calculate rolling resistance

### Definition

Rolling resistance is the resistive force applied to the wheel when it’s rolling. The magnitude of rolling resistance depends on a series of factors, which most of the time act together and can not be broken down clearly.

Most common causes of rolling resistance applied to a wheel in motion are:

• tire deformation
• friction between tire and road surface
• wheel friction with surrounding air
• friction in the wheel hub and bearings

In the following discussion we are going to focus on the rolling resistance caused by the tire and road. The wheel hub and bearings efficiency is not part of this lecture since they can be consider as part of the overall driveline losses (efficiency).

Image: Tire pressure distribution – stationary vs. rolling

In the image above: N [N] – normal reaction force, W [N] – weight force acting on the wheel and ω [rad/s] – angular speed of the wheel.

When the tire is stationary, the distribution of the normal force N [N] in the contact patch is symmetric with respect with the vertical axis of the wheel. However, when the wheel is in motion (rotating), the normal force distribution is not anymore symmetric but concentrates in the front of the wheel, towards the direction of motion. This makes the resultant force N [N] to create a resistive rolling torque, which will try to slow down the wheel.

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### Formula

Image: Rolling resistance – wheel forces diagram

Let’s assume we have a vehicle which has front-wheel drive (FWD). The rear wheels are going to be pulled along, through the chassis. Therefore there will be a pulling force Fp [N] acting in the wheel hub, which will force the wheel to rotate. As explained previously, due to asymmetric force distribution in the contact patch, there is going to be a normal force N [N] acting on the tire, at the distance a [m] from the vertical axis of the tire.

The force N [N] is the vertical component of a resultant force going through the tire’s centre of rotation. The horizontal component of that resultant force, acting in the contact patch, which is trying to slow down the tire is exactly the rolling resistance force Frr [N].

Since the wheel is in equilibrium, the sum of forces on the x-axis, the sum of forces on the y-axis and the sums of torques acting around the centre of the wheels are all zero.

• x-axis forces equilibrium
ΣFx = 0
(1)
Fp – Frr = 0
(2)
Fp = Frr
(3)
• y-axis forces equilibrium
ΣFy = 0
(4)
N – W = 0
(5)
N = W
(6)
• torques equilibrium
ΣT = 0
(7)
Frr · rw – N · a = 0
(8)

Replacing N from (6) in (8) and solving for Frr [N] gives:

Frr = (a/rw) · W
(9)

The ratio between distance a [m] and wheel radius rw [m] is the rolling resistance coefficient f [-].

f= a/rw
(10)

Replacing (10) in equation (9) gives the general formula of the rolling resistance force for flat (no gradient) roads.

Frr = f · W = f · m · g
(11)

where m [kg] is the vehicle mass and g = 9.81 m/s2 is the gravitational acceleration.

If we calculate the total rolling resistance force of the vehicle, then the entire vehicle mass is used in the equation. If we calculate the rolling resistance of only one wheel, then the vehicle mass is divided by four (assuming equal weight distribution between wheels).

If the vehicle is rolling on a road with the gradient α [°], then the formula for rolling resistance becomes:

Frr = f · W · cos(α) = f · m · g · cos(α)
(12)

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### Rolling resistance coefficient

The rolling resistance coefficient of a tire depends on tire construction, materials, air pressure, vehicle speed, and road conditions. In general, for low vehicle speeds, the value of rolling resistance coefficient is constant.

The rolling resistance coefficient slightly increases with the increase of the traveling speed of the vehicle. According to [7], under normal operating conditions, with the vehicle speed below 200 kph, the rolling resistance coefficient is between 0.01 – 0.02.

There are several sources of automotive literature, in which you can find the value of the rolling resistance coefficient, function on several parameters. For example in [6], you can find the value of the rolling resistance coefficient function of the road type.

 Road surface Rolling resistance coefficient Pneumatic car tires on large set pavement 0.013 small set pavement 0.013 concrete, asphalt 0.011 rolled gravel 0.02 tarmacadam 0.025 unpaved road 0.05 field 0.1 – 0.35 Pneumatic truck tires on concrete, asphalt 0.006 – 0.01 Strake wheels in field 0.14 – 0.24 Track-type tractor in field 0.07 – 0.12 Wheel on rail 0.001 – 0.002

Table: Coefficients of rolling resistance
Source: [6]

From [1] we can also extract a table of rolling resistance coefficients function of road surface.

 Road surface Rolling resistance coefficient new, firm asphalt; concrete; small pavement; cobblestone pavement 0.005 – 0.015 rolled, firm gravel; wear down, washboard asphalt 0.02 – 0.03 tarred, wear down, washboard gravel 0.03 – 0.04 very well dirt roads 0.04 – 0.05 dirt roads 0.05 – 0.15 sand 0.15 – 0.35

Table: Coefficients of rolling resistance
Source: [1]

Additional values for the rolling resistance coefficient can be found in [8] for different pavement types at low vehicle speeds.

 Pavement type Rolling resistance coefficient Good asphalt or concrete pavement 0.01 – 0.018 General asphalt or concrete pavement 0.018 – 0.02 Gravel road 0.02 – 0.025 Good gravel road 0.025 – 0.030 Pebble potholes pavement 0.035 – 0.050 Pressed dirt road (dry) 0.025 – 0.035 Pressed dirt road (rainy) 0.050 – 0.150 Muddy dirt road 0.100 – 0.250 Dry sand 0.100 – 0.300 Wet sand 0.060 – 0.150 Icy roads 0.015 – 0.030 Compacted ski track 0.030 – 0.050

Table: Rolling resistance coefficient of a vehicle at low speed on a given road
Source: [8]

In reality, the rolling resistance coefficient depends on several factors, like:

• tire construction
• slip between tire and road
• inflating pressure

In the image below you can see the influence of tire pressure on the rolling resistance coefficient. As the air pressure increases, the rolling resistance coefficient decreases. This happens because an increased internal pressure leads to a further stiffening of the tire, which means that the tire deflection decreases at steady state load. This leads to a decrease of the tire flexing energy and due to the smaller contact patch, to a decline of the frictional component of resistance [1].

Image: Rolling resistance coefficient function of wheel load and tire pressure
Credit: [1]

In the next image you can see the influence of vehicle speed on the rolling resistance. The increase of the rolling resistance coefficient, which rises with the vehicle speed is due to the superimposed effect of the tire deformation wave on the flexion resistance. This effect increases with velocity [1].

Image: Rolling resistance coefficient function of driving speed
Credit: [1]

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### Models for rolling resistance coefficient

For calculation/simulation purposes, you can use either a constant value of the rolling resistant coefficient or a speed dependant one. In the automotive literature, there are a number of formulas for the rolling resistance coefficient, mainly function of vehicle speed and tire pressure.

Rolling resistance coefficient function of vehicle speed

In [9] there are several proposed formulas for the rolling resistance coefficient.

$f (V)= f_{0} + f_{01} \cdot V + f_{02} \cdot V^{2} \tag{13}$

where V [kph] is vehicle speed and f0, f01 and f02 are coefficients function of tire construction. For example, for a radial tire, the coefficients can be set as:

f0 = 1 · 10-2
f01 = 5 · 10-7
f02 = 2 · 10-7

If we plot f [-] function of V [kph], we get the following curve:

Image: Rolling resistance coefficient (1)

The coefficients can be adjusted in order to obtain the desired behaviour of the rolling resistance coefficient. For a road made from good asphalt or concrete, a rolling coefficient which varies between 0.01 at 0 kph and 0.025 at 250 kph is good enough for simulations.

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### Models for rolling resistance force

Once we have the rolling resistance coefficient function of vehicle speed, we can apply equation (12) and get the rolling resistance force.

For example, if we consider that the total mass of the vehicle is 2000 kg and that the weight is evenly distributed on all four wheels, we’ll get a weight on a wheel of:

Frr = (2000/4) · 9.81 · f(V)
(14)

Applying equation (12) and the rolling resistance coefficient defined by (13), we can plot the rolling resistance force function of vehicle speed.

Image: Rolling resistance force function of speed

In the SAE standard J2452, the rolling resistance force is defined function of vehicle speed, tire inflation pressure and the load applied on the wheel.

Frr = Zα · Pβ · (a + b · V + c · V2)
(15)

where:

Frr [N / lbs] – rolling resistance force
P [kPa / psi] – tire inflation pressure
Z [N / lbs] – tire load
V [kph / mph] – vehicle speed
α, β, a, b, c [-] – tire dependant coefficients

As an example, from [3] we can extract the coefficients for the tire T40 of size 205/75R15 97S:

α = 1.03399904
β = -0.41081927
a = 0.05933157
b = 9.85526e-05
c = 3.72314e-07

For the same tire load of 4905 N and different inflation pressures of 150, 250 and 350 kPa, we can plot the rolling resistance using equation (15):

Image: Rolling resistance force function of speed pressure and load

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### Example

For a vehicle of mass 1800 kg, travelling on a road with the slope angle of 10° and with a fixed rolling resistance coefficient of 0.012, calculate the total rolling resistance force.

Applying equation (12), we get:

Frr = 1800 · 9.81 · 0.012 · cos(10 · π / 180) = 209 N

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### Calculator

 m [kg] = f [-] = α [°] = Frr [N] =

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### References

[1] Henning Wallentowitz, Longitudinal Dynamics of Vehicles – Lecture, IKA RWTH, Aachen, 2004.
[2] Lars Eriksson, Lars Nielsen, Modeling and Control of Engines and Drivelines, Wiley, 2014.
[3] Ben Wen, Gregory Rogerson, and Alan Hartke, Correlation Analysis of Rolling Resistance Test Results from SAE J1269 and J2452, SAE International, 2014.
[4] Green Seal’s Report, Low Rolling Resistance Tires, March, 2003.
[5] Transportation Research Board, Tires and Passenger Vehicle Fuel Economy, Transportation Research Board Special Report 286, 2006.
[6] Automotive Handbook, 9th Edition, Bosch, 2014.
[7] Georg Rill, Vehicle Dynamics – Lecture Notes, University of Applied Sciences, Regensburg, 2005.
[8] Wuwei Chen et al, Integrated Vehicle Dynamics and Control, Wiley, 2016.
[9] M. Untaru et al, Dinamica Autovehiculelor pe Roti, Editura Didactica si Pedagogica, Bucuresti, 1981.