Rail Adhesion - Directional Stability and Hunting Instability

Directional Stability and Hunting Instability

While common belief dictates that the wheels are kept on the tracks by the flanges, the flanges in reality make few contacts 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 heat but also including noise and, if sustained, would lead to excessive wheel wear.

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. It should be noted that some railway systems employ a flat wheel and track profile, relying on cant alone to reduce flange contact, e.g. Melbourne suburban network, Australia.

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. This problem was alleviated to a great extent by ensuring the diameter of all coupled wheels was very closely matched.

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. It is necessary 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:

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. Since the wavelength increases with reducing taper, 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:

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.