Definitions

# Rail adhesion

The term adhesion railway or adhesion traction describes the most common type of railway, where power is applied by driving some or all of the wheels of the locomotive and thus it relies on the friction between a steel wheel and a steel rail.

The term is particularly used when discussing mountain railways to distinguish from other forms of traction such as funicular or cog railway (rack and pinion). For example, the Bernese Oberland Railway "is a mixed rack and adhesion railway"

Traction or friction can be reduced when the rails are greasy, due to rain, oil, or decomposing leaves which compact into a hard slippery lignin coating. On an adhesion railway most locomotives have a sandbox containing sand which can be sprayed on to the rail to improve traction under slippery conditions.

Measures against reduced adhesion due to leaves include application of 'Sandite' (a gel-sand mix) by special sanding trains, scrubbers and water jets, and long-term management of railside vegetation.

The UK media commonly ridicule rail operators who give leaves on the line as an explanation for railway delays.

## Effect of Adhesion Limits

The process of adhesion is complex, possibly consisting of a combination of the effects of small-scale roughness of the two surfaces and the attraction between the molecules near the surface of the two faces in contact. Whatever the underlying cause, the effects can be represented quite simply. Usually, we only need to predict when sliding occurs, or, if the vehicle relies on sliding for its operation, how large the propulsive force must be.

Usually the force needed to start sliding is greater than that needed to continue sliding. The former is concerned with static friction, referred colloquially to as 'stiction', or more pretentiously as 'limiting friction', whilst the latter is called 'sliding friction'. At what speed the situation ceases to be 'static' and becomes 'sliding' is never defined, as the process of transition between the two is not particularly well understood.

Experience shows that the heavier an object is, the harder it is to drag. Newton's Second Law dictates that we should expect the acceleration of the heavy object to be less, because it has the greater inertia, but in the absence of friction, some motion is to be expected. So our simple model of static friction defines a resisting force parallel to the surface, which is proportional to the force perpendicular to the surface, the object will not move until the applied motive force exceeds this resisting force. The constant of proportionality is called the coefficient of friction. Values for typical engineering materials are given in the Engineer's Handbook

When a wheel rolls, the point of contact is stationary with respect to the rail, so under normal conditions, an adhesion railway is governed by static friction. For steel on steel the coefficient of friction can be as high as 0.78, under the best of conditions, whilst under extreme conditions it can fall to as low as 0.05. A 100 tonne locomotive could have a tractive effort of 78 tonnes , under the ideal conditions (assuming the power can be matched to the load), falling to a mere 5 tonnes under the worst conditions.

A well turned (railway) wheel, in good bearings should achieve a lift to drag ratio as high as 50:1, so the locomotive would require 2 tonnes of tractive effort just to move itself, leaving just enough to pull two 60-tonne coaches, provided the train doesn't encounter a gradient. More typically, we should expect the coefficient of friction to be in the region of 0.25, implying that the 100 tonne locomotive could pull 21 coaches on the level. Alternatively, the same locomotive could pull ten coaches up a 12% gradient.

Safety, however, would dictate that railway brakes should still be effective in the worst adhesion conditions, so gradients would normally be restricted to roughly 5% or less, unless measures are taken to ensure adequate adhesion under all circumstances.

This restriction on the gradient which can be used introduces a requirement for tunnels, cuttings, embankments and bridges to keep the track level. A further restriction is introduced by the radius of turn.

Since railway wheels usually have flanges for safety, the radius of turn is not directly determined by the coefficient of friction. Ultimately, the turn performance will be limited by toppling. This will occur when the overturning moment due to the side force (centrifugal or centripetal acceleration) is sufficient to cause the inner wheel to begin to lift off the rail. Once this occurs, the ability of the vehicle to resist further toppling reduces, so that unless the side force is reduced, the vehicle will topple completely.

For a wheel gauge of 1.5 m, centre of gravity height of 3 m and speed of 30 m/s (108 km/h), the radius of turn is 360 m. For a modern high speed train at 80 m/s, the toppling limit would be about 2.5 km. In practice, the minimum radius of turn is much greater than this, as contact between the wheel flanges and rail at high speed could cause significant damage to both. For very high speed, the minimum adhesion limit again appears appropriate, implying a radius of turn of about 13 km. In fact, the outer rail would be super-elevated to tilt the train into the bend, so that the practical turn limit is closer to 7 km.

The constraints on gradient and radius of turn impose severe limitations on the choice of routes. This problem is compounded in developed countries by the difficulty of acquiring the land needed for new routes. Evidently, legacy track designed to accommodate steam traction is unlikely to be adequate for the current generation of high speed train.

High tractive effort requires a heavy locomotive, which is ideally suited to the low power to weight ratio of steam engines. The weight has two adverse effects: firstly, the many bridges needed to keep the track level must be designed to withstand the weight of the locomotive; and secondly, the bearing load of the engine on the track would be high. Heavy steam locomotives designed in the 20th Century had typically between 6 and 10 large diameter drive wheels in order to distribute the load, but these represent the result of a long evolution.

During the 19th Century, it was widely believed that coupling the drive wheels would compromise performance, and was avoided on engines intended for express passenger service. With a single drive wheelset, the Herzian contact stress between the wheel and rail necessitated the largest diameter wheels that could be accommodated. The weight of locomotive was restricted by the stress on the rail, and sandboxes were required, even under reasonable adhesion conditions.

## Directional Stability and Hunting Instability

What keeps the train on the track? The common answer is the wheel flanges. Actually, the flanges rarely make contact with the track, and when they do, most of the contact is sliding. The rubbing of a flange on the track dissipates large amounts of energy, mainly as noise, and if sustained would lead to wheel wear. Evidently, throwing energy away by rubbing flanges against the rail is not the best way to achieve low rolling losses.

Close examination of a typical railway wheel reveals that the tread is burnished, but the flange is not; the flanges rarely make contact with the rail. The tread of the wheel is slightly tapered. When the train is in the centre of the track, the region of the wheels in contact with the rail traces out a circle which has the same diameter for both wheels. The velocities of the two wheels are equal, so the train moves in a straight line.

If, however, the wheelset is displaced to one side, the diameters of the regions of contact, and hence the (linear) velocities of the wheels, are different, and the wheelset tends to steer back towards the centre. Also, when the train encounters a bend, the wheelset displaces laterally slightly, so that the outer wheel speeds up (linearly) and the inner wheel slows down, causing the train to turn the corner.

Understanding how the train stays on the track, it becomes evident why Victorian locomotive engineers were averse to coupling wheelsets. This simple coning action is possible only with wheelsets where each can have some free motion about its vertical axis. If wheelsets are rigidly coupled together, this motion is restricted, so that coupling the wheels would be expected to introduce sliding, resulting in increased rolling losses.

With perfect rolling contact between the wheel and rail, this coning behaviour manifests itself as a swaying of the train from side to side. In practice, the swaying is damped out below a critical speed, but is amplified by the forward motion of the train above the critical speed. This lateral swaying is known as 'hunting'. The phenomenon of hunting was known by the end of the 19th Century, although the cause was not fully understood until the 1920s, and measures to eliminate it were not taken until the late 1960s. As is often the case, the limitation on maximum speed was imposed not by raw power, but by encountering an instability in the motion.

The kinematic description of the motion of tapered treads on the two rails is insufficient to describe hunting well enough to predict the critical speed. We need to deal with the forces involved. There are two phenomena which must be taken into account. The first is the inertia of the wheelsets and vehicle bodies, giving rise to forces proportional to acceleration; the second is the distortion of the wheel and track at the point of contact, giving rise to elastic forces. The kinematic approximation corresponds to the case which is dominated by contact forces.

A fairly straightforward analysis of the kinematics of the coning action yields an estimate of the wavelength of the lateral oscillation:

$lambda=frac\left\{1\right\}\left\{2pi\right\}sqrt\left\{frac\left\{rd\right\}\left\{2k\right\}\right\}$

where d is the wheel gauge, r is the nominal wheel radius and k is the taper of the treads. For a given speed, the longer the wavelength and the lower the inertial forces will be, so the more likely it is that the oscillation will be damped out. Now the wavelength increases with reducing taper, so increasing the critical speed requires the taper to be reduced, which implies a large minimum radius of turn.

A more complete analysis, taking account of the actual forces acting, yields the following result for the critical speed of a wheelset:

$V^2=frac\left\{Wrad^2\right\}\left\{k left\left(4C+md^2 right\right)\right\}$

where W is the axle load for the wheelset, a is a shape factor related to the amount of wear on the wheel and rail, C is the moment of inertia of the wheelset perpendicular to the axle, m is the wheelset mass.

The result is consistent with the kinematic result in that the critical speed depends inversely on the taper. It also implies that the weight of the rotating mass should be minimised compared with the weight of the vehicle. The wheel gauge implicitly appears in both the numerator and denominator, implying that it has only a second-order effect on the critical speed.

The true situation is much more complicated, as the response of the vehicle suspension must be taken into account. Restraining springs, opposing the yaw motion of the wheelset, and similar restraints on bogies, may be used to raise the critical speed further. However, in order to achieve the highest speeds without encountering instability, a significant reduction in wheel taper is necessary, so there is little prospect of reducing the turn radius of high speed trains much below the current value of 7 km.

## Forces on Wheels

The behaviour of adhesion railways is determined by the forces arising between two surfaces in contact. This may appear trivially simple from a superficial glance, but it becomes extremely complex when studied to the depth necessary to predict useful results.

The first error to address is the assumption that wheels are round. A glance at a parked car will immediately show that this is not true; the region in contact with the road is noticeably flattened, so that the wheel and road conform to each other over a region of contact. If this were not the case, the contact stress of a load being transferred through a point contact would be infinite. Rails, and railway wheels, are much stiffer than pneumatic tyres and tarmac, but the same distortion takes place at the region of contact. Typically, the area of contact is elliptical, of the order of 15 mm across.

The distortion is small and localised, but the forces which arise from it are large. In addition to the distortion due to the weight, both wheel and rail distort when braking and accelerating forces are applied, and when the vehicle is subjected to side forces. These tangential forces cause distortion in the region where they first come into contact, followed by a region of slippage. The net result is that during traction the wheel does not advance as far as would be expected from rolling contact, but during braking it advances further. This mix of elastic distortion and local slipping is known as "creep" (not to be confused with the creep of materials under constant load). The definition of creep in this context is:

$mbox\left\{creep\right\}=frac\left\{\left(mbox\left\{actual displacement\right\} - mbox\left\{rolling displacement\right\}\right)\right\}\left\{\left(mbox\left\{rolling displacement\right\}\right)\right\} ,$

In analysing the dynamics of wheelsets, and complete rail vehicles, the contact forces are treated as linearly dependent on the creep.

The forces which result in directional stability, propulsion and braking may all be traced to creep. It is present in a single wheelset, and will accommodate the slight kinematic incompatibility introduced by coupling wheelsets together, without causing gross slippage, as was once feared.

Provided the radius of turn is sufficiently great (as should be expected for express passenger services), two or three linked wheelsets should not present a problem. However, 10 drive wheels (5 main wheelsets) are usually associated with heavy freight locomotives.

## References

• Sir Charles Inglis: Applied Mathematics for Engineers pp194,195, Cambridge University Press, 1951
• Carter F W: On the Stability of Running of Locomotives, Proc. Royal Society, July 25 1928
• Wickens A H: The Dynamics of Railway Vehicles on Straight Track: Fundamental Considerations of Lateral Stability, Proc. Inst. Mech. Eng. 1965-66,p29
• Wickens A H, Gilchrist A O and A E W Hobbs: Suspension Design for High-Performance Two-Axle Freight Vehicles, Proc. Inst. Mech. Eng. 1969-70, p22

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