The push a millimetre ahead of the axle

The force that takes what it needs found that static friction has no value of its own and supplies whatever equilibrium demands up to a limit. The grip that needs a little slipping found that a tyre delivering a force is never purely rolling: part of its contact patch is stuck and part slides, and the force it transmits is a measure of the sliding. The contact that slips before it slides followed that partial slip into a contact that is not rolling at all and found it losing energy in a ring at the edge long before the whole contact gives way. All three were about contacts asked to transmit a sideways force.

This essay takes the sideways force away. A wheel rolling freely on a flat road, with nothing driving it and nothing braking it, should go on for ever: no surface slides over another, and the contact point is momentarily at rest. It does not go on for ever. It slows, and keeping it at constant speed takes a steady push of about a hundredth of the load for a car tyre, a thousandth for a train wheel. Calling that push “rolling friction” suggests a coefficient like the sliding one and a cause like the sliding one. Neither is right. The push is the torque of a force applied in the wrong place.

Where the ground pushes #

A wheel carrying a load $W$ is held up by the ground with a force equal to $W$. If the ground pushes directly below the axle, the push passes through the centre and exerts no torque, and the wheel can roll at constant speed with nothing else acting. If the ground pushes at a point a distance $d$ ahead of the axle, the push has a torque $W d$ about the centre, trying to turn the wheel backwards. To keep it rolling steadily something must supply an equal and opposite torque, and if it is a horizontal force $F$ at the axle, acting at height $R$ above the contact, then $FR = Wd$:

$$\frac{F}{W} = \frac{d}{R}.$$

The rolling-resistance coefficient is a ratio of two lengths: how far ahead the push acts, over how big the wheel is. Coulomb measured it in 1781 by rolling wooden cylinders on wooden tracks and found the force proportional to the load and inversely proportional to the radius, which is exactly what a fixed offset implies. He called the offset a property of the materials. What he could not say was why the ground pushes ahead of the axle at all, since a wheel pressing on a perfectly elastic surface is pushed symmetrically, as much from behind the contact as in front of it.

A surface that answers late #

The answer is that no real surface is perfectly elastic. A piece of road, or of the rubber in a tyre, is compressed as the contact arrives and released as it leaves, and the stress it develops depends not only on how far it has been squeezed but on how fast. Squeezed quickly, it resists more than its elastic stiffness alone would give; released, it recovers more slowly than the wheel moves away. The material ahead of the axle, being loaded, pushes hard; the material behind, unloading, pushes less and in the extreme case lets go of the wheel before the wheel has left it.

The figure computes this for a rigid wheel rolling over a bed of independent elements, each behaving like a rubber-like material with two relaxation times, 2 and 20 milliseconds: slowly loaded it has one stiffness, quickly loaded ten times more. For each speed the depth to which the wheel sinks is found by requiring the bed to carry the load.

At a millimetre a second every element has time to relax completely as the wheel passes, the bed behaves as a soft elastic solid, and the pressure is symmetric about the axle: no torque, no resistance. At a thousand metres a second no element has time to relax at all, the bed behaves as a stiff elastic solid, the wheel sinks less, and the pressure is symmetric again. At a metre a second the pressure is lopsided. It rises steeply at the front, where the elements are being squeezed at a rate they cannot accommodate, and falls away well before the rear of the contact, where the recovering elements have not caught up with the wheel and have stopped touching it. The resultant sits 6.4 millimetres ahead of the axle, on a wheel 300 millimetres in radius, and the coefficient is 0.021.

Nothing slides at any point. Every element of the bed moves straight down and straight up, and the wheel’s rim meets it without relative motion. The resistance is entirely the lopsidedness of a vertical push.

The speed that costs the most #

Sweeping the speed shows the whole story in one curve per load. The resistance is zero at a crawl and at very high speed and peaks between. The peak sits where the time a piece of the bed spends under the wheel — the length of the contact divided by the speed — is about equal to the material’s relaxation time. That is the same condition that makes a material’s loss tangent largest: the width that is a lifetime found that an oscillator absorbs most at the frequency of its own decay, and a viscoelastic solid, loaded once as a wheel passes, absorbs most when the loading takes about as long as its internal rearrangements. A heavier load makes a longer contact, which takes longer to cross, and so moves the peak to a higher speed and makes it taller.

The energy view says the same thing in other words. A rolling wheel deforms each piece of the bed once and releases it, and the energy stored in the deformation is returned only in part. The fraction lost per cycle, divided into the elastic energy stored per metre rolled, is the drag force. A material that loses nothing returns everything and has no drag; a material with a large loss at the frequency the wheel imposes has a large one. The rubber’s loss is the mechanical counterpart of the constant that depends on how fast it is asked, the dielectric that stores or dissipates according to whether its molecules can turn in time. Rolling resistance is hysteresis, read through the geometry of a contact.

Big wheels, light loads #

The lever-arm picture explains at once why big wheels roll more easily. The offset $d$ is a fraction of the contact’s length, and a bigger wheel makes a longer contact, but only slowly: on this bed the contact grows about as the cube root of the radius, while the lever it acts against grows as the radius itself. So the coefficient falls roughly as $R^{-2/3}$. A heavier load makes a longer contact too, growing as about the cube root of the load, so the coefficient rises as roughly $W^{1/3}$ and the drag force itself as $W^{4/3}$. The same powers come out of Tabor’s estimate for an elastic ball rolling on a material that loses a fixed fraction of its strain energy, using Hertz’s theory for the size of the contact.

That is why carts that had to cross soft ground were built with tall wheels, why a pram with large wheels is easier to push than a shopping trolley with small ones, and why the wheels of luggage that rolls smoothly on a station floor stop on a carpet: the carpet’s fibres are a bed with large losses at walking pace, and a small wheel sinks into them relative to its radius. On sand and soft soil the loss is not viscoelastic but plastic — the ground is compacted and stays compacted — and the coefficients are tens of times larger, but the geometry is the same: a wheel climbing continuously out of the hollow it makes.

Millimetres, everywhere #

Read as lever arms, the familiar rolling-resistance coefficients become small distances. A car tyre on asphalt, with a coefficient of about a hundredth and a radius of 310 millimetres, is pushed about three millimetres ahead of its axle. A bicycle’s narrow, high-pressure tyre is pushed a millimetre and a third ahead; a railway wheel, steel on steel, about two-thirds of a millimetre. Every number is set by the material and how much of it flexes. A tyre’s rubber flexes over a contact patch the size of a hand and through the whole of its sidewall on every revolution; a steel wheel’s contact patch is the size of a coin and its steel loses a tiny fraction of the energy it stores. That is the whole reason a train needs a fraction of the energy per tonne-kilometre that a lorry does, and why railways replaced canals rather than roads.

It also explains what tyre pressure does. A soft tyre makes a longer, deeper contact and flexes more of its rubber further on every turn, so its offset grows; inflating it shortens the contact and shrinks the offset. The relation is roughly that the coefficient falls as the square root of the pressure, and it is one of the few ways a driver can change the resistance of a tyre already fitted.

The slope a wheel will stand on #

The lever arm has a consequence that anyone who has parked on a slight incline has met. A wheel resting on a slope is pulled downhill by the component of its weight along the slope, which about the contact point has a torque of $W R \sin\theta$. Nothing slides, so friction in the ordinary sense holds the contact point; whether the wheel rolls depends on whether the ground’s push can shift far enough uphill of the contact to balance that torque. The push can move within the contact patch, by up to about the same offset it takes when rolling, and so a wheel stays put on any slope with $\tan\theta$ smaller than $d/R$.

For a car tyre that is a slope of about a hundredth, half a degree: a car left out of gear with its brakes off on a road that falls a centimetre in a metre may not move at all, and needs a push to start. A marble on a thick carpet sits still on a slope of several degrees, because its tiny radius and the carpet’s deep, lossy fibres make $d/R$ large. A steel ball on glass rolls on a slope of a few thousandths of a degree. The threshold is the rolling counterpart of the force that takes what it needs: up to the limit, the offset of the push adjusts to whatever is needed, and the wheel shows no sign of being under any strain.

Once it does roll, the same offset is what limits how fast a rolling body can speed up on a slope. The mass and where it sits found that a hoop and a disc race down a slope at different rates because each must spin up its own moment of inertia, with no energy lost on the way. Rolling resistance adds a steady loss on top, and on a shallow enough slope it wins, and a body that would roll on glass stops on felt.

Grip and drag are one loss at two frequencies #

The surprising part is that the property that makes a tyre roll badly is the same property that makes it grip. A tyre’s grip on a wet road comes largely from hysteresis too: as the rubber slides slowly over the bumps of the road’s texture, it is deformed and released by each bump, and the energy it loses in doing so is a force resisting the slide. The bumps that matter are a fraction of a millimetre to a few millimetres apart, and a tyre skidding at a few metres a second meets them at thousands to millions of times a second. Rolling, by contrast, deforms each part of the tread about ten times a second, once per revolution.

So a tyre engineer wants rubber with a large loss at high frequencies, for grip, and a small loss at low frequencies, for low rolling resistance. Rubber’s loss depends on frequency and temperature together, as the response of a liquid that remembers depends on how fast it is asked to flow — warming a rubber shifts all its relaxation times as cooling and slowing would — and the compounds used in low-resistance tyres since the 1990s, in which silica replaces much of the carbon black that had always filled tread rubber, were designed to shape that loss curve: high near the frequencies of skidding at road temperature, low near the frequency of rolling. The figure’s peak, moved to another speed, is the same peak. The grip that is not a coefficient found that sliding friction between rough elastic bodies has a geometric origin; on rubber, much of it has this viscoelastic one, and it cannot be separated from what the same rubber costs when it rolls.

What is lost, and where it goes #

The energy a rolling wheel loses becomes heat in the material that flexed. A car tyre on a motorway runs tens of degrees warmer than the air, and that warming feeds back: warmer rubber has its relaxation times shortened, which moves its loss peak and changes its resistance, and the steady temperature of a tyre on a long run is set by the balance between the heat its hysteresis generates and what the air and the road carry away. A rolling-resistance coefficient is therefore measured at a specified speed, load, pressure and temperature, and published values describe tyres warmed to steady running. The fuel spent overcoming rolling resistance in a car at moderate speeds is of the order of a fifth to a third of the total, the rest going mainly into air drag, which overtakes it at around eighty kilometres an hour.

The wheel itself gives back nothing it has lost, and in that respect rolling resembles the bounce an elastic plate cannot give back, where a perfectly elastic collision still lost energy into vibrations. There the energy went into waves that left; here it goes into a material that does not answer in time. In both, a quantity that looks like a material constant — a coefficient of restitution, a coefficient of rolling resistance — turns out to be a property of how fast the motion is compared with how fast the material can respond.

What the pictures cannot show #

The computed bed is a model: independent elements with no coupling between neighbours, a single pair of relaxation times, a rigid wheel and a load given per metre of width. A real tyre is the soft partner, not the road, and its losses are spread through its tread, its sidewalls and its internal plies, with the contact patch flattened against a stiff road rather than sunk into a soft one; its rubber has a broad spread of relaxation times rather than two, so its loss peak is broader. The figures ignore adhesion between the surfaces, which adds a small resistance as they are peeled apart behind the contact, and microslip, which adds another when the wheel transmits any force. The lever arms in the last figure come from typical published coefficients and radii, which vary by tens of per cent with pressure, temperature and surface.

Still open: predicting a tyre’s loss before it is made #

A tyre’s rolling resistance can be measured on a drum in a few hours, and it can be estimated from the measured loss properties of its compounds with finite-element models of the whole tyre, which agree with measurement to within about ten per cent. What cannot yet be done reliably is to predict the loss properties of a new rubber compound from its recipe — how the filler particles, the polymer chains and the bonds between them will together set the loss at each frequency and temperature — which is why compounds are still developed largely by making and testing them. How far the trade between grip and rolling resistance can be pushed by structuring a rubber at the scale of its filler particles is an active question for an industry whose products account for a sizeable share of every road vehicle’s fuel use.

The habit worth carrying away is to ask where a force acts as well as how large it is. A rolling wheel is resisted by no sliding at all, but by the ground’s push acting a millimetre or so ahead of its axle — the lag of a material that answers late — so the coefficient is a lever arm over a radius, d/R, peaking when the time under the wheel matches the material’s relaxation time and falling for bigger wheels. A railway wheel’s push is two-thirds of a millimetre ahead, a car tyre’s three, and the same loss that costs a tyre its fuel buys it its grip.