Rolling resistance force
The most usual way to represent the tyre energetic losses is rolling resistance force that acts in the wheel axis. Its value multiplied by the distance covered by the tyre yields the work value consumed for the displacement and hence the total energy lost in the rolling process. Starting from the general free-body tyre diagram presented in figure 1, there are some alternative representations of rolling forces and moments (figure 2).
Figure 1 - Free body tyre diagram
Figure 2 - Alternative representation of rolling resistance forces and moments
The value of the rolling resistance force is influenced by:
- Tyre operating conditions;
- Tyre construction.
The analysis of the influence factors effects is complex, taking into consideration the fact that they are more or less interconnected. This is the reason why this analysis will further target general principles rather than numerical values. Also, it must be mentioned the impressive work done by D.J. Schuring  that was used as guide for following presentation.
Before to discuss the specific effects of the operating condition, first must be stated the major role of the tyre temperature. So, the increasing of load or speed is accompanied by an increase in heat production and hence, in rolling resistance (figure 3). Because the heat transfer to the environment increases slower than heat generation, a gradual increase of tyre temperature appears. Higher temperature will reduce the hysteresis losses and increase the inflation pressure. Due to these effects the rolling resistance decrease (i.e. for a 385/65 R 22.5 tyre with 25 kN load and a final speed of 60 km/h, the inflation pressure increases from 600 to 678 kPa and the rolling resistance force decreases from 172 to 127 N ). If the operating conditions remain stable, these evolutions continues until an equilibrium state is reached (the heat generation become equal to the hear transfer). The time necessary to reach this equilibrium ranges from 100 to 150 min for truck tyres.
Figure 3 - Effect of load/speed changes on tyre temperature, inflation and rolling loss
The rolling resistance measurements are usually performed in two ways:
- At constant temperature - the data are taken immediately after a change has been made and all the parameter except those under investigation remain constant;
- At equilibrium temperature - the data are taken after the temperature equilibrium has been reached.
In some procedures the pressure changes are suppressed by connecting the tyre to a pressure regulator. Tests in which pressure is kept constant are named "regulated", while the others "capped".
The main operating conditions influence factors are:
- Rolling speed - In the case of measurements carried out at an equilibrium temperature on a capped tyre, a certain reduction in the rolling resistance is noticed with slower running speeds. The explanation lies in the stronger influence of thermal factors, which diminish hysteresis losses and increase tyre pressure rather than the dynamic factors, which cause its increase. In the range of moderate speeds (40 to 80 km/h) the rolling resistance is kept approximately constant or increases slightly in a linear manner (figure 4), while in the range of high speeds they start to increase rapidly. This is accounted for by the very high (developing) inertia forces. These inertia forces cause the restitution of an ever-smaller amount from the elastic deformation energy, inducing also the vibration of the tyre tread (stationary waves). The energy of the vibration is then consumed via the hysteresis of the material. This trend is more evident on passenger car tyres than on larger truck tyres. The explication consists in the much higher stiffness of truck tyres construction and inflation pressure.
In the case of measurements done at equilibrium temperature with inflation pressure kept constant, it could be observed higher speed dependence, the rolling resistance increasing all the time.
In the third case, at constant temperature when data are taken immediately after change, appears the most pronounced increasing with speed (figure 5).
Figure 4 - Rolling resistance force vs. speed (at equilibrium temperature, capped), adapted from 
Figure 5 - Rolling resistance force vs. speed (at constant temperature, regulated), adapted from 
- Vertical load - Starting from the observation that in most situations, the rolling resistance is almost proportional to the vertical load, researchers have defined the ratio of these values as the rolling resistance coefficient. Actually, when measuring at the equilibrium temperature, this coefficient is almost constant or slightly decreasing for radial truck tyres (figure 6). An explanation of these variations could be that a load increase leads to an increase of the deformation and thus, of the energy losses. On the other hand, due to the rise effect of the equilibrium temperature together with the load rise, an effect of reduction of losses appears. Finally, these two contradictory trends yield the variation of one value with respect to the other.
For constant temperature measurements, the rolling resistance coefficient seems to be almost constant, the rolling resistance force increasing linear with load (figure 7). However, the increasing rates of rolling resistance force are higher under constant temperature conditions than under equilibrium (figure 8).
Figure 6 - Rolling resistance coefficient vs. load (at equilibrium temperature), adapted from 
Figure 7 - Rolling resistance force vs. load (at constant temperature), adapted from 
Figure 8 - Compare of rolling resistance force increasing rates, adapted from 
- Inflation pressure - When referring to the tyre air pressure, most researchers actually mean the initial pressure value corresponding to cold tyre, as it is well known that the pressure rises due to the temperature rise during rolling. However, in some papers, one may present results with respect to the pressure recorded during the equilibrium temperature. The experimental results show a decrease in the rolling resistance with the inflation pressure increase. The variation of the rolling resistance depending on tyre air pressure under the equilibrium temperature, for a radial truck tyre, is given in figure 9.
Figure 9 - Rolling resistance force vs. inflation pressure (at equilibrium temperature), adapted from 
- Environment temperature - The research carried out and presented in the professional literature, conclude that one cannot make accurate measurements without taking into account the environment temperature. Thus, it is noticed that the rolling resistance drops by 0.5 to 1 % with a temperature increase of 1oC .
- Roadway surface type - An important element defining roadway type is its rugosity. Here I mean the irregularities that are smaller than the contact area. The influence of this factor obtained on different roads with different types of coating is given in figure 10. . The surface 1 (new concrete with relatively smooth macro-texture), was taken as a reference. The roadway profile noted 2 and 3 (rolled asphalt composed of mixed aggregate) represent the extremes of typical primary public highways. Although the action mechanism (hysteresis or slide friction) is not fully comprehended, the rugosity of the roadway has no negligible effect upon the rolling resistance.
A special problem arises with rolling on wet or snow laid roadways. On the wet roads the rolling resistance for a tyre running at 60 km/h may rise by approximate 10 % with respect to the one determined for the dry surface. This rise is accountable for by the cooling effect exerted by water, by the change in the deformation of the tread and by the energy required for water displacement.
Figure 10 - Road surface influence on rolling resistance 
- Truck axles geometry - In order to obtain the lowest rolling resistance, the geometry of all axles of a truck (and trailer) must have a properly alignment. Improper wheel toe-in and camber angles or misalignment of front axle, drive and trailer tandem axles (figure 11) could conduct to significant increases on rolling losses.
Figure 11 - Alignment/disalignment of truck axle
The main tyre construction influence factors are:
- Carcass and belt design - The radial designed truck tyres shows lower rolling resistance compared with bias designed tyres (up to 30 % differences). The main explication is the fact that the radial tyres undergo diminished strains as a result of more flexible carcass and lower share stresses between plies. The plies number of the carcass has also an important influence to the rolling losses. For example, if the plies number increases from 3 to 6, the losses increase with approximate 5 % .
Referring to the cord wires, we must state their dual influence. First they
absorb an amount of deformation energy due to their hysteresis phenomenon, and second, they control the deformation amplitude (and thus the amount of losses in the neighborhood rubber). In spite of the fact that in the literature some specialists credited the loss caused by cord wires themselves with 20 - 80 % of the tyre total loss, an estimation performed by calculation using real data  state the conclusion that this loss are much lower (approximate 5 %). As a result of research, it could be said that influence of the cord wire section dimension is negligible while the influence of the between cords rubber lies is major.
- Tread design - Using a dual compound configuration, called a cap/base tread (figure 12), the rolling resistance could be improved. While the cap region is made of rubber with improved treadwear and adherence characteristics, the base of the tread is made of a rubber with low loss characteristics. This construction can reach an improvement of 5 % over the single compound tread .
Figure 12 - Dual compound tread
- Tyre geometry - Taking into account the tyre transversal section dimensions, some of them could affect significant the rolling resistance.
- Rim width - The major part of researchers agree that, in a particular case of a radial tyre, the minimum value of the rolling resistance is obtained for an optimum ratio between the width of the rim and the width of the tyre.
- Tyre exterior radius - It is known that a large tyre has lower rolling resistance than a small tyre. The explication consists in the fact that the exterior radius is in close connection with the tyre nominal load. Thus, a large tyre has higher stiffness and encounters lower deformations over the same vertical load.
- Tyre cross section ratio (ratio between height and width of the tyre cross section) - For the radial tyres, a part of researchers considers that the rolling resistance decreases with the decrease of this ratio as a result of the tyre stiffness increase while an other part considers that exist an optimum related to the specific conditions. In spite of this controversy, for a given load, a wide base tyre has lower rolling resistance than a pair of dual conventional tyres with the same load rating because of less rubber that is engaged ,  (figure 13).
Figure 13 - Wide base tyre vs. dual assembly of conventionl tyres
- The tread geometry - Considering the large volume of rubber of the tread, its geometry affect significant the energetic losses. So, the decrease of the tread thickness could reduce the rolling resistance up to 30 % (reduction of 1 mm reduce the rolling resistance of a heavy truck tyre by 3 %) . More, some researchers state a linear dependence between them. As a result, a practical conclusion is that
the rolling resistance diminishes with the tread wear. The tread design
aggressiveness that can be evaluated as the net-to-gross contact area
ratio, influence also the truck tyre rolling resistance. So, a decrease
from 100 to 60 % could increase the rolling resistance with approximate 25 % . The tread cross section curvature has a significant influence. A small curvature radius implies large tread deformation, but on the other hand, diminishes strains of the tyre shoulder. Due to these two opposite effects it seems to be an optimum for each specific case.
- Tyre materials - the main materials used in tyre construction are those for the cord wires and the rubber compounds.
- The cord material type - it has a reduced influence on rolling losses in case of the carcass (less of 6 % differences) and somewhat higher in case of the belt.
- The rubber type and composition - The tyre compound parts are made of different types of rubber or rubber mixtures in order to provide an optimum of performances. The rubber composition has the major influence on rolling losses due to the hysteresis phenomenon. As was mentioned earlier the hysteresis curve has a variation with temperature
as is shown before. The energetic losses are influenced by the right side hysteresis curve values, which must be to a low level.
Finally, it must remember that in many cases the rolling resistance is presented as a
rolling resistance coefficient obtained from the rolling resistance force divided with the tyre vertical load force. Typical values for truck tyres are 0.005 - 0.007 for speeds between 20 and 80 km/h.