# Rolling resistance

The rolling resistance (also: rolling friction or rolling friction ) is the force that arises when a wheel or rolling element rolls and is directed against the movement . Since the rolling resistance is roughly proportional to the normal force , the rolling resistance coefficient (also: rolling resistance coefficient , rolling friction coefficient , etc.) is formed as a characteristic value . ${\ displaystyle c _ {\ mathrm {R}}}$

${\ displaystyle F _ {\ mathrm {R}} = c _ {\ mathrm {R}} \ cdot F _ {\ mathrm {N}}}$ , Rolling resistance is the coefficient of rolling resistance times normal force

The values ​​for the rolling resistance coefficients are considerably smaller compared to the appropriate values ​​for sliding friction . Rolling bearings (e.g. ball bearings ) therefore have advantages over plain bearings in terms of friction.

The rolling resistance coefficient depends not only on the material pairing but also on the geometry (radius of the rolling element). ${\ displaystyle c _ {\ mathrm {R}}}$

The force that has to be overcome in order to bring a body (for example a wheel) from a standstill into rotating motion is known as the starting resistance .

## Basics

Asymmetrical contact force
Forces while rolling

When rolling, both the rolling body and the base (the roadway ) are deformed. The deformation takes place both on the rolling element itself and on the rolling element path, namely at the point of contact or at the contact line. Although this is essentially an elastic deformation, there are also processes that cause a loss of energy. These are, for example, sliding friction components when rolling off the off-center tire parts or when cornering, flexing in the tire rubber or plastic processes in the ground (driving on sand, grit or processes in the gravel of the track bed). Examples:

• Combination steel wheel - rail on the railway. In the picture opposite, the rail surface is elastically deformed by the wheel; when moving, the rail material is compressed in the direction of travel. A mountain piles up in front of the bike. Since the rail material moves only slightly, the mountain is continuously rolled through the wheel and leveled again behind the wheel. When slipping, the material is strongly pressed due to the high surface pressure . The more often a rail is used, the more likely it is that parts of the surface will break out as a result of the pressure and relaxation, which can be identified by matt or rough surfaces. A side effect is the combination of rails and sleepers lying in a flexible ballast bed, which has a dampening effect. Since the wheel is in the "valley" of the impression point while driving, it has to be constantly recreated, even on a horizontal route. It moves with you while driving, which means a corresponding loss of energy. However, the effect is smaller than the above. In addition, there is friction when cornering due to the rigid axles.
• Even if a rubber-tired vehicle is on a soft surface, e.g. B. loose sand, the driving becomes more difficult the narrower the tires are, i. H. the deeper they sink into the sand. All-terrain vehicles such as mountain bikes etc. therefore tend to have wide tires.
• Combination of tire and road surface , in which, conversely, the deformation of the tire is usually visibly greater than that of the (paved) road. An increased air pressure in the tire reduces the rolling resistance because it reduces the flexing work and the contact area. This also makes cornering easier, but has negative effects on road holding and driving comfort, so that a compromise is usually necessary. Even with bicycles, the rolling resistance is on a hard surface, e.g. B. Asphalt, the less, the harder you inflate the tires. Bicycles for solid surfaces, e.g. B. racing bikes , therefore have rather narrow tires and drive with a similarly high internal tire pressure as trucks.
• Rolling so-called medicine balls requires extra strength. This is achieved by the fact that these have a filling that is not only relatively heavy, but also plastic within certain limits , which is constantly deformed when unrolled and requires additional energy.
• Often, as a result of the permanent load, material fatigue occurs and parts are detached. This is the so-called pitting (English: pitting ).

## Rolling resistance coefficient

Due to the deformation when rolling, the contact force between the body and the surface becomes asymmetrical (Fig. 1). The replacement of the contact forces by statically equivalent individual forces results in a normal force F N , which is shifted forward by the distance d , and a friction force F R against the direction of movement (Fig. 2).

From the equilibrium conditions it follows for rollers with radius R at constant speed

${\ displaystyle F _ {\ mathrm {R}} = {\ frac {d} {R}} \ cdot F _ {\ mathrm {N}}}$

The quotient is the rolling resistance coefficient c R (also outdated: rolling resistance coefficient. Rolling friction coefficient ): ${\ displaystyle {\ frac {d} {R}}}$

${\ displaystyle c _ {\ mathrm {R}} = {\ frac {d} {R}}}$

This gives the expression for the rolling friction F R the form

${\ displaystyle F _ {\ mathrm {R}} = c _ {\ mathrm {R}} \ cdot F _ {\ mathrm {N}}}$

Meaning: "arm" of the normal force, the radius of the wheel : normal force${\ displaystyle d}$${\ displaystyle R:}$${\ displaystyle F _ {\ mathrm {N}}}$

The rolling resistance coefficient is a dimensionless (unit-free) number that depends on the material properties and the geometry of the rolling body (in the case of tires, this depends on the air pressure). Typical numerical values ​​of the rolling resistance coefficient are one to over two orders of magnitude below those of the sliding friction coefficient .

### Typical rolling resistance coefficients c R

c R Rolling elements / rolling element track
0.0005-0.001 Hardened steel ball bearing , ball and bearing 4
0.001-0.002 Railway wheel on rail 1
0.015-0.02 Motorcycle tires on asphalt
0.006-0.010 Car tires on asphalt, truck
0.011-0.015 Car tires on asphalt, cars
0.01-0.02 Car tires on concrete 2
0.02 Car tires on gravel
0.015-0.03 Car tires on cobblestones 2
0.03-0.06 Car tires on pothole track 2
0.045 Crawler track ( track drive , Leopard 2 tank ) on slab track
0.050 Tires on earth away
0.04-0.08 Car tires on stuck sand 2
0.035-0.08 Webbing (crawler tracks, Caterpillar Challenger and John Deere 8000T) on asphalt
0.2-0.4 Car tires on loose sand 2 , 3
1 Gustav Niemann are for railway wheels following (determined from tests) formula to: ; d and D in mm. With a wheel diameter of 800 mm, this results in approx. 0.4 mm, which corresponds to a coefficient of 0.001.${\ displaystyle d = 0 {,} 013 \ cdot {\ sqrt {D}}}$
2 Source: Schmidt, Schlender 2003
3 Anyone who has ever tried to ride a bike on the beach can confirm these high numerical values ​​from their own experience

### Limits of Theory

The relationship described above is a simplified model that is sufficient for most calculations in the art. The dependence of the friction on other variables (contact force, speed, etc.) is not taken into account (see also breakaway resistance ). ${\ displaystyle F _ {\ mathrm {R}} = {\ frac {d} {R}} \ cdot F _ {\ mathrm {N}} = c _ {\ mathrm {R}} \ cdot F _ {\ mathrm {N} }}$

Furthermore, the model described does not consider the possible influence of a third substance that may be present at the boundary layer between the rolling element and the rolling element path (liquid or lubricant). Examples are grease on the rails or water on the road. In such a case, the term mixed friction is used.

Extreme values ​​for speeds and temperatures as well as possible chemical influences at the contact points cannot be recorded with this model.

## literature

• Valentin L. Popov: Contact Mechanics and Friction. A text and application book from nanotribology to numerical simulation. Springer-Verlag, 2009, ISBN 978-3-540-88836-9 .