Rolling resistance

Figure 1  Hard wheel rolling on and deforming a soft surface, resulting in the reaction force R from the surface having a component that opposes the motion. (W is some vertical load on the axle, F is some towing force applied to the axle, r is the wheel radius, and both friction with the ground and friction at the axle are assumed to be negligible and so are not shown. The wheel is rolling to the left at constant speed.) Note that R is the resultant force from non-uniform pressure at the wheel-roadbed contact surface. This pressure is greater towards the front of the wheel due to hysteresis.

Rolling resistance, sometimes called rolling friction or rolling drag, is the resistance that occurs when a round object such as a ball, tire, or wheel rolls on a surface. It is caused mainly by the deformation of the object, the deformation of the surface, and movement below the surface. Additional contributing factors include wheel radius, forward speed load on wheel, surface adhesion, sliding, and relative micro-sliding between the surfaces of contact. It depends very much on the material of the wheel or tire and the sort of ground. What might be termed "basic rolling resistance" is steady velocity and straight line motion on a level surface, but there also exists rolling resistance when accelerating, when on curves, and when on a grade.

For example, rubber will give a bigger rolling resistance than steel on some surfaces (polished steel) and a lower rolling resistance on other surfaces (pavement/tarmac). Also, sand on the ground will give more rolling resistance than concrete. Any moving wheeled vehicle will gradually slow down due to rolling resistance including that of the bearings, but a train car with steel wheels running on steel rails will roll farther than a bus of the same mass with rubber tires running on tarmac. The coefficient of rolling resistance is generally much smaller for tires or balls than the coefficient of sliding friction.

Primary cause

The primary cause of rolling resistance is hysteresis:

A characteristic of a deformable material such that the energy of deformation is greater than the energy of recovery. The rubber compound in a tire exhibits hysteresis. As the tire rotates under the weight of the vehicle, it experiences repeated cycles of deformation and recovery, and it dissipates the hysteresis energy loss as heat. Hysteresis is the main cause of energy loss associated with rolling resistance and is attributed to the viscoelastic characteristics of the rubber.

-- National Academy of Sciences

Thus materials that flex more and bounce back slowly, such as rubber, exhibit more rolling resistance than materials that flex less, such as steel, or that bounce back more quickly, such as silica. Low rolling resistance tires typically incorporate silica in place of carbon black in their tread compounds to reduce low-frequency hysteresis without compromising traction.

Railroads also have hysteresis in the roadbed structure and in the pure rolling resistnace

Factors that contribute in tires

Several factors affect the magnitude of rolling resistance a tire generates:

• As mentioned in the introduction: wheel radius, forward speed, surface adhesion, and relative micro-sliding.
• Material - different fillers and polymers in tire composition can improve traction while reducing hysteresis. The replacement of some carbon black with higher-priced silica–silane is one common way of reducing rolling resistance. The use of exotic materials including nano-clay has been shown to reduce rolling resistance in high performance rubber tires.^{[citation needed]} Solvents may also be used to swell solid tires, decreasing the rolling resistance.
• Dimensions - rolling resistance in tires is related to the flex of sidewalls and the contact area of the tire For example, at the same pressure, wider bicycle tires flex less in sidewalls as they roll and thus have lower rolling resistance (although higher air resistance).
• Extent of inflation - Lower pressure in tires results in more flexing of sidewalls and higher rolling resistance. This energy conversion in the sidewalls increases resistance and can also lead to overheating and may have played a part in the infamous Ford Explorer rollover accidents.
• Over inflating tires (such a bicycle tires) may not lower the overall rolling resistance as the tire may skip and hop over the road surface. Traction is sacrificed, and overall rolling friction may not be reduced as the wheel rotational speed changes and slippage increases.^{[citation needed]}
• Sidewall deflection is not a direct measurement of rolling friction. A high quality tire with a high quality (and supple) casing will allow for more flex per energy loss than a cheap tire with a stiff sidewall.^{[citation needed]} Again, on a bicycle, a quality tire with a supple casing will still roll easier than a cheap tire with a stiff casing. Similarly, as noted by Goodyear truck tires, a tire with a "fuel saving" casing will benefit the fuel economy through many tread lives (i.e. retreading), while a tire with a "fuel saving" tread design will only benefit until the tread wears down.
• In tires, tread thickness and shape has much to do with rolling resistance. The thicker and more contoured the tread, the higher the rolling resistance Thus, the "fastest" bicycle tires have very little tread and heavy duty trucks get the best fuel economy as the tire tread wears out.
• Smaller wheels, all else being equal, tend to have higher rolling resistance than larger wheels^{[citation needed]}. However, in some laboratory tests, smaller wheels appeared to have similar or lower losses than large wheels, but these tests were done rolling the wheels against a small-diameter drum, which would theoretically remove the advantage of large-diameter wheels, thus making the tests irrelevant for resolving this issue. Another counter example to the general rule of smaller wheels have higher rolling resistance can be found in the area of ultimate speed soap box derby racing. In this race, the speeds have increased as wheel diameters have decreased by up to 50%. This suggests that rolling resistance may not be increasing significantly with smaller diameter within a practical range.
• Virtually all world speed records have been set on relatively narrow wheels,^{[citation needed]} probably because of their aerodynamic advantage at high speed, which is much less important at normal speeds.
• Temperature: with both solid and pneumatic tires, rolling resistance has been found to decrease as temperature increases (within a range of temperatures: i.e. there is an upper limit to this effect)

Rolling resistance coefficient

The "rolling resistance coefficient", is defined by the following equation:

$\ F = C_{rr} N$

where

$F$ is the rolling resistance force (shown in figure 1),

$C_{rr}$ is the dimensionless rolling resistance coefficient or coefficient of rolling friction (CRF), and

$N$ is the normal force the weight of the vehicle on the wheel (including the weight of the wheel)

$C_{rr}$ is just the pounds of force needed to push (or tow) a wheeled vechicle forward (at constant speed on the level with no air resistance) weighing one pound. It's assumed that all wheels are the same and bear identical weight. Thus :$\ C_{rr} = 0.01$ means that it would only take 0.01 pound to tow a vehicle weighing one pound. For a 1000 pound vehicle it would take 1000 times more tow force or 10 pounds. $C_{rr}$ is in lb(tow-force)/lb(vehicle weight). Since this lb/lb is force divided by force, $C_{rr}$ is dimensionless. Multiply it by 100 and you get the percent of the weight of the vehicle required to maintain slow steady speed. $C_{rr}$ is often multiplied by 1000 resulting in a unit of kilograms (kg force) per metric ton (tonne = 1000 kg ) which is the same as pounds of resistance per 1000 pounds of load or Newtons/kilo-Newton, etc. For the US railroads, lb/ton has been traditionally used which is just $2000 C_{rr}$. Thus they are all just measures of resistance per unit vehicle weight. While they are all "specific resistances" sometimes they are just called "resistance" although they are really a coefficient (ratio)or a multiple thereof. If using force units like pounds or kilograms, mass is equal to weight so one could claim that $C_{rr}$ is also the force per unit mass (only true on Earth with uniform gravity). The SI system would use N/tonne (N/T) which is $1000 g C_{rr}$ where g is the acceleration of gravity in SI units (meters per second).

The above shows resistance proportional to $C_{rr}$ but does not explicitly show any variation with speed, loads, surface roughness, diameter, tire inflation/wear, etc. because $C_{rr}$ itself varies with those factors. It might seem from the above that the rolling resistance is directly proportional to vehicle weight but that's not so for railroad vehicles.

Measurement

There are at least two popular models for calculating rolling resistance.

1. "Rolling resistance coefficient (RRC). The value of the rolling resistance force divided by the wheel load. The Society of Automotive Engineers (SAE) has developed test practices to measure the RRC of tires. These tests (SAE J1269 and SAE J2452) are usually performed^{[citation needed]} on new tires. When measured by using these standard test practices, most new passenger tires have reported RRCs ranging from 0.007 to 0.014." In the case of bicycle tires, values of 0.0025 to 0.005 are achieved. These coefficients are measured on rollers, with power meters on road surfaces, or with coast-down tests. In the latter two cases, the effect of air resistance must be subtracted or the tests performed at very low speeds.
2. The coefficient of rolling resistance b, which has the dimension of length, is approximately (due to the small-angle approximation of $cos(\theta) = 1$) equal to the value of the rolling resistance force times the radius of the wheel divided by the wheel load.
3. ISO 8767 is used to test rolling resistance in Europe.

The results of these tests can be hard for the general public to obtain as manufacturers prefer to publicize "comfort" and "performance".

Physical formulas

The coefficient of rolling friction for a slow rigid wheel on a perfectly elastic surface, not adjusted for velocity, can be calculated by

$\ C_{rr} = \sqrt {z/d}$

where

$z$ is the sinkage depth

$d$ is the diameter of the rigid wheel

The force of rolling resistance can also be calculated by:

$\ F = \frac{N b}{r}$

where

$F$ is the rolling resistance force (shown in figure 1),

$r$ is the wheel radius,

$b$ is the rolling resistance coefficient or coefficient of rolling friction with dimension of length, and

$N$ is the normal force (equal to W, not R, as shown in figure 1).

The above equation, where resistance is inversely proportional to radius r. seems to be based on the discredited "Coulumb's law". See #Diameter of wheel. Equating this equation with the force per the #Rolling resistance coefficient, and solving for b, gives b = C_rr·r. Therefore, if a source gives rolling resistance coefficient (C_rr) as a dimensionless coefficient, it can be converted to b, having units of length, by multiplying C_rr by wheel radius r.

Rolling resistance coefficient examples

Table of rolling resistance coefficient examples: [2]

Crr	b	Description
0.0019 to 0.0065		Mine car cast iron wheels on steel rail
0.0010 to 0.0024	0.5 mm	Railroad steel wheel on steel rail. Passenger rail car about 0.0020
0.0003 to 0.0004		"Pure rolling resistance" Railroad steel wheel on steel rail
0.001 to 0.0015	0.1 mm	Hardened steel ball bearings on steel
0.0022 to 0.005		production bicycle tires at 120 psi (8.3 bar) and 50 km/h (31 mph), measured on rollers
0.0025		Special Michelin solar car/eco-marathon tires
0.005		Dirty tram rails (standard) with straights and curves[citation needed]
0.0055		Typical BMX bicycle tires used for solar cars
0.0062 to 0.015		Car tire measurements
0.010 to 0.015		Ordinary car tires on concrete
0.3		Ordinary car tires on sand
0.0045 to 0.008		Large truck (Semi) tires
0.0385 to 0.073		Stage coach (19th century) on dirt road. Soft snow on road for worst case.

For example, in earth gravity, a car of 1000 kg on asphalt will need a force of around 100 newtons for rolling (1000 kg × 9.81 m/s² × 0.01 = 98.1 N).

Diameter of wheel

Per Dupuit (1837) rolling resistance (of wheeled carriages) is approximately inversly proportional to the square root of wheel diameter. This rule has been experimentally verified for cast iron wheels (8" - 24" diameter) on steel rail and for 19th century carriage wheels. But there are other tests of carriage wheels that don't agree. Theory of a cylinder rolling on an elastic roadway also gives this same rule These contradict earlier (1785) tests by Coulomb of rolling wooden cylinders where Coulomb reported that rolling resistance was inversely proportional to the diameter of the wheel (known as "Coulomb's law"). . But this disputed (or wrongly applied) -"Coulomb's law" is still found in handbooks.

"Rolling resistance" can mean different things

In the broad sense, specific "rolling resistance" (for vehicles) is the force per unit ton of vehicle weight required to move the vehicle on level ground and a constant slow speed where aerodynamic drag (air resistance) is insignificant and also where there are no traction (motor) forces or brakes applied. In other words the vehicle would be coasting if it were not for the force to maintain constant speed. An example of such usage for railroads is [3].

The pure "rolling resistance" for a train is that which happens due to deformation and possible minor sliding at the wheel-road contact. . For a rubber tire, the energy loss happens over the entire tire, but it's still called "rolling resistance". In the broad sense, "rolling resistance" includes bearing resistance, energy loss by shaking both the roadbed (and the earth underneath) and the vehicle itself, and by sliding of the wheel, road/rail contact. Railroad textbooks seem to cover all these resistance forces but do not call their sum "rolling resistance" (broad sense) as is done in this article. They just sum up all the resistance forces (including aerodynamic drag) and call the sum basic train resistance (or the like).

Since railroad rolling resistance in the broad sense may be a few times larger than just the pure rolling resistance reported values may be in serious conflict since they may be based on different definitions of "rolling resistance". The trains engines must of course, provide the energy to overcome this broad-sense rolling resistance.

Sound and heat effects

Rolling friction generates heat and sound (vibrational) energy, as mechanical energy is converted to these forms of energy due to the friction. One of the most common examples of rolling friction is the movement of motor vehicle tires on a roadway, a process which generates sound and heat as by-products. The sound generated by automobile and truck tires as they roll (especially noticeable at highway speeds) is mostly due to the percussion of the tire treads, and compression (and subsequent decompression) of air temporarily captured within the treads. The heat generated raises the temperature of the frictional surface; moreover, this temperature increase typically increases the coefficient of friction. This is why automobile racing teams preheat their tires.

Comparing rolling resistance of highway vehicles and trains

While the specific rolling resistance of a train is far less than an automobile or truck in terms of resistance force per ton, this doesn't necessarily means that the resistance force per passenger or per net ton of freight is less. It all depends on the weight of the vehicle per passenger or per net ton transported. Thus one needs to know the rolling resistance per passenger (or per net ton) to make such comparisons.

Weight per passenger

For 1975, Amtrak passenger trains weighed a little over 7 tones per passenger while automobiles weighed only a little over one ton per passenger. To find the rolling resistance per person one multiples the pounds(force) per ton (2000 times the rolling resistance coefficient) by the tons per passenger. This means that even if the rolling coefficient is several times greater for the auto than for the train, then after multiplication to get pounds/passenger, there is not a lot of difference between the two values (of lb/passenger). Thus there may not be a large difference in the rolling resistance energy used to transport a person by rail as compared to auto.

Gross tons per net ton

For U.S. rail freight in 1975 there was about 2.5 gross tons of train (including cargo) for every ton of freight cargo. What was it for trucks? It was likely lower but it appears that the low rolling resistance of a rail wheel gives rail a big advantage in lower rolling resistance overall. However, for some very lightweight (low density) goods, the heavy weight of the rail cars and locomotives could give trucks a lower rolling resistance per revenue ton than rail. See Rail vs. Truck Energy Efficiency

See also

• Coefficient of friction
• Low-rolling resistance tires
• Maglev (Magnetic Levitation, the elimination of rolling and thus rolling resistance)
• Rolling element bearing

References

• Астахов П.Н. (Russian) "Сопротивление движению железнодорожного подвижного состава" (Resistance to motion of railway rolling stock) Труды ЦНИИ МПС. Выпуск 311. - М.: Транспорт, 1966. – 178 pp.

(In 2012, full text was on the Internet but the U.S. was blocked)

• Деев В.В., Ильин Г.А., Афонин Г.С. (Russian) "Тяга поездов" (Traction of trains)

Учебное пособие. - М.: Транспорт, 1987. - 264 pp.

• Hay, William W. "Railroad Engineering" New York, Wiley 1953
• Hersey, Mayo D., "Rolling Friction" Transactions of the ASME, April 1969 pp. 260-275 and Journal of Lubrication Technology, January 1970, pp. 83-88 (one article split between 2 journals) Except for the "Historical Introduction" it's mainly about lab. testing of mine railroad cast iron wheels of diameters 8" to 24".
• Hoerner, Sighard F., "Fluid dynamic drag", published by the author, 1965. (Chapt. 12 is "Land-Borne Vehicles" and includes rolling resistance (trains, autos, trucks).
• Roberts, G. B., "Power wastage in tires", International Rubber Conference, Washington, D.C. 1959.
• U.S National Bureau of Standards, "Mechanics of Pneumatic Tires", Monograph #132, 1969-1970.
• Williams J A, "Engineering tribology" Oxford University Press, 1994.

External links

• physics tutorial
• temperature vs rolling resistance
• Simple roll-down test to measure Crr in cars and bikes