Hunting oscillation

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The classical Hunting oscillation is a swaying motion of a railway vehicle caused by the coning action on which the directional stability of an adhesion railway depends.

Below a certain critical speed, the motion is damped out, above this speed the motion can be violent, damaging track and wheels, and potentially causing derailment.

It was discovered towards the end of the 19th Century, when speeds became high enough to encounter it. Remedial measures, particularly in the design of suspension systems have been introduced since the 1960s, permitting speeds exceeding 180mph (290km/h).

It arises from the interaction of adhesion forces and inertial forces. At low speed adhesion dominates, but as the speed increases the adhesion forces and inertial forces become comparable in magnitude, and the oscillation begins at the critical speed.

1 Classical Wheelset Hunting
- 1.1 Kinematic Analysis
- 1.2 Energy Balance
2 Concluding Comments
3 References
4 See also

[edit] Classical Wheelset Hunting

[edit] Kinematic Analysis

Deeper understanding of the phenomenon inevitably requires a mathematical analysis of the vehicle dynamics, which may not be accessible to all readers.

A kinematic description deals with the geometry of motion, without reference to the forces causing it, so the analysis begins with a description of the geometry of a wheel set running on a straight track. Since Newton's Second Law relates forces to accelerations of bodies, the forces acting may then be derived from the kinematics by calculating the accelerations of the components.

The train stays on the track by virtue of the conical shape of the wheel treads. If a wheelset is displaced to one side by an amount y, the radius of the tread in contact with the rail on one side is reduced, whilst on the other side it is increased. The angular velocity is the same for both wheels (they are coupled via a rigid axle), so the larger diameter tread speeds up, whilst the smaller slows down. The wheel set steers around a centre of curvature defined by the intersection of the generator of a cone passing through the points of contact with the wheels on the rails and the axis of the wheel set. Applying similar triangles, we have for the turn radius:

$\frac {1} {R} =\frac {2ky}{rd}$

where d is the track gauge, r the wheel radius when running straight and k is the tread taper.

The path of the wheel set relative to the straight track is defined by a function y(x) where x is the progress along the track. Provided the direction of motion remains more or less parallel to the rails, the curvature of the path may be related to the second derivative of y with respect to distance along the track:

$\frac {d^2y} {dx^2}=\frac {1}{R}$

It follows that the trajectory along the track is governed by the equation:

$\frac {d^2y} {dx^2}=-(\frac{2k}{rd}) y$

This is a simple harmonic motion having wavelength:

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

This kinematic analysis implies that trains sway from side to side all the time. In fact this oscillation is damped out below a critical speed, and the ride is correspondingly more comfortable. The kinematic result ignores the forces causing the motion. These are difficult to quantify simply, as they arise from the elastic distortion of the wheel and rail at the regions of contact, and the subsequent motion requires analysis which lies outside the scope of this article (see ref.1).

If the motion is substantially parallel with the rails, the angular displacement of the wheel set ( $θ$ ) is given by:

$\theta=\frac {dy}{dx}$

Hence:

$\frac{d \theta}{dx}=\frac{d^2y}{dx^2}=-(\frac{2k}{rd}) y$

$\frac{d^2 \theta}{dx^2} = -(\frac{2k}{rd}) \frac {dy}{dx} = -(\frac{2k}{rd}) \theta$

The angular deflection also follows a simple harmonic motion, which lags behind the side to side motion by a quarter of a cycle. In many systems which are characterised by harmonic motion involving two different states (in this case the axle yaw deflection and the lateral displacement), the quarter cycle lag between the two motions endows the system with the ability to extract energy from the forward motion. This effect is observed in 'flutter' of aircraft wings and 'shimmy' of road vehicles, as well as hunting of railway vehicles. The kinematic solution derived above describes the motion at the critical speed.

In practice, the lag between the two motions is less than a quarter cycle below the critical speed, so the motion is damped out, but is greater than a quarter cycle above the critical speed, and the motion is amplified.

In order to estimate the inertial forces, it is necessary to express the distance derivatives as time derivatives. This is done using the speed of the vehicle U, which is assumed constant:

$\frac {d }{dt} = U\frac {d }{dx}$

The angular acceleration of the axle in yaw is:

$\frac {d^2 \theta}{dt^2} = -U^2 (\frac{2k}{rd}) \theta$

The inertial moment (ignoring gyroscopic effects) is:

$Fd = C \frac{d^2 \theta}{dt^2}$

where F is the force acting along the rails and C is the moment of inertia of the wheel set.

$F= -C U^2 (\frac{2k}{rd^2}) \theta$

the maximum frictional force between the wheel and rail is given by:

$F=\mu \frac {W}{2}$

where W is the axle load and $μ$ is the coefficient of friction. Gross slipping will occur at a combination of speed and axle deflection given by:

$\theta U ^2 = \mu W \frac{rd^2}{(4Ck)}$

this expression yields a significant overestimate of the critical speed, but it does illustrate the physical reason why hunting occurs, i.e. the inertial forces become comparable with the adhesion forces above a certain speed. Limiting friction is a poor representation of the adhesion force in this case.

The actual adhesion forces arise from the distortion of the tread and rail in the region of contact. There is no gross slippage, just elastic distortion and some local slipping. During normal operation these forces are well within the limiting friction constraint. A complete analysis takes these forces into account.

[edit] Energy Balance

In order to get an estimate of the critical speed, we use the fact that the condition for which this kinematic solution is valid corresponds to the case where there is no net energy exchange with the surroundings, so by considering the kinetic and potential energy of the system, we should be able to derive the critical speed. Let:

$\omega = \frac{d\theta}{dt}$

Using the operator:

$\omega \frac {d}{d\theta} = \frac{d}{dt}$

the angular acceleration equation may be expressed in terms of the angular velocity in yaw : $ω$

$\omega \frac {d \omega}{d\theta} = -U^2 (\frac{2k}{rd}) \theta$

integrating:

$\frac {1}{2} \omega^2=-1/2 U^2 (\frac{2k}{rd}) \theta^2$

so the kinetic energy due to rotation is:

$\frac{1}{2} C \omega^2 = -1/2 C U^2 (\frac{2k}{rd}) \theta^2$

When the axle yaws, the points of contact move outwards on the treads so that the height of the axle is lowered. The distance between the support points increases to:

$\frac{d}{cos(\theta)} = d(1+\frac{1}{2} \theta^2)$

(to second order of small quantities). the displacement of the support point out from the centres of the treads is:

$\frac{1}{2} (d+\frac{d \theta^2}{2}-d)$

the axle load falls by

$h=\frac{kd \theta^2}{4}$

The work done by lowering the axle load is therefore:

$E=\frac{Wkd \theta^2}{4}$

This is energy lost from the system, so in order for the motion to continue, an equal amount of energy must be extracted from the forward motion of the wheelset.

The outer wheel velocity is given by:

$V = \frac{U (r+ky)}{r}$

The kinetic energy is:

$\frac { m (U^2 + 2U^2 \frac {ky}{r}+U^2 \frac {k^2 y^2}{r^2})}{4}$

for the inner wheel it is

$\frac { m (U^2 - 2U^2 \frac{ky}{r}+U^2\frac {k^2 y^2}{r^2})}{4}$

where m is the mass of both wheels.

The increase in kinetic energy is:

$\delta E = \frac {1}{2}\frac {mU^2 k^2}{r^2} y^2$

The motion will continue at constant amplitude as long as the energy extracted from the forward motion, and manifesting itself as increased kinetic energy of the wheel set at zero yaw, is equal to the potential energy lost by the lowering of the axle load at maximum yaw.

Now, from the kinematics: $2U\frac{ky}{(rd)} = \omega$

$\delta E =\frac {1}{8} m d^2 \omega^2$

but

$\omega^2=- U^2 (\frac{2k}{rd}) \theta^2$

The translational kinetic energy is

$\delta E = -\frac {1}{8} U^2 md^2 (\frac {2k}{rd}) \theta^2$

The total kinetic energy is:

$T=\frac{1}{2} U^2(C+ \frac {md^2}{4}) (\frac {2k}{rd}) \theta^2$

The critical speed is found from the energy balance:

$\frac{Wkd}{2}=U^2 (\frac {2k(C+\frac {md^2}{4})}{rd})$

Hence the critical speed is given by

$U^2=\frac{Wrd^2}{(4C+md^2)}$

This is independent of the wheel taper, but depends on the ratio of the axle load to wheel set mass. If the treads were truly conical in shape, the critical speed would be independent of the taper. In practice, wear on the wheel causes the taper to vary across the tread width, so that the value of taper used to determine the potential energy is different from that used to calculate the kinetic energy. Denoting the former as a, the critical speed becomes:

$U^2=\frac{Ward^2}{(k(4C+md^2)}$

where a is now a shape factor determined by the wheel wear. This result is derived in reference 2 from an analysis of the system dynamics using standard control engineering methods.

[edit] Concluding Comments

The motion of a wheel set is much more complicated than this analysis would indicate. There are additional restraining forces applied by the vehicle suspension, and at high speed, the wheel set will generate additional gyroscopic torques, which will modify the estimate of the critical speed. A real railway vehicle has many more degrees of freedom, and consequently may have more than one critical speed, and it is by no means certain that the lowest is dictated by the wheelset motion.

However, the analysis is instructive because it shows why hunting occurs. As the speed increases the inertial forces become comparable with the adhesion forces. That is why the critical speed depends on the ratio of the axle load (which determines the adhesion force) to the wheelset mass (which determines the inertial forces).

Alternatively, below a certain speed, the energy which is extracted from the forward motion is insufficient to replace the energy lost by lowering the axles, and the motion damps out, above this speed, the energy extracted is greater than the loss in potential energy, and the amplitude builds up.

The potential energy at maximum axle yaw may be increased by including an elastic constraint on the yaw motion of the axle, so that there is a contribution arising from spring tension. Arranging wheels in bogies to increase the constraint on the yaw motion of wheelsets, and applying elastic constraints to the bogie also raises the critical speed. Introducing elastic forces into the equation permits suspension designs which are limited only by the onset of gross slippage, rather than classical hunting. The penalty to be paid for the virtual elimination of hunting is a straight track, with an attendant right of way problem, and incompatibility with legacy infrastructure.

Hunting is a dynamic problem which can be solved, in principle at least, by active feedback control, which may be adapted to the quality of track. However, the introduction of active control raises reliability and safety issues.

Shortly after the onset of hunting, gross slippage occurs and the wheel flanges impact on the rails, potentially causing damage to both.