Hunting oscillation

Self oscillation about an equilibrium that is usually unwanted

Hunting oscillation is a self oscillation, usually unwanted, about an equilibrium. The expression came into use in the 19th century and describes how a system "hunts" for equilibrium. The problem was first noticed towards the end of the 19th century, when train speeds became high enough to encounter it. Serious efforts to counteract it got underway in the 1930s, giving rise to lengthened trucks and the side damping swing hanger truck. In the development of the Japanese Shinkansen, less conical wheels and other design changes were used to extend truck design speeds above Advances in wheel and truck design based on research and development efforts in Europe and Japan have extended the speeds of steel wheel systems well beyond those attained by the original Shinkansen, while the advantage of backwards compatibility keeps such technology dominant over alternatives such as the hovertrain and maglev systems. The speed record for steel wheeled trains is held by the French TGV, at .

While a qualitative description provides some understanding of the phenomenon, deeper understanding inevitably requires a mathematical analysis of the vehicle dynamics. Even then, the results may be only approximate. 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 the acceleration of bodies, the forces acting may then be derived from the kinematics by calculating the accelerations of the components. However, if these forces change the kinematic description (as they do in this case) then the results may only be approximately correct.

This kinematic description makes a number of simplifying assumptions since it neglects forces. For one, it assumes that the rolling resistance is zero. A wheelset (not attached to a train or truck), is given a push forward on a straight and level track. The wheelset starts coasting and never slows down since there are no forces (except downward forces on the wheelset to make it adhere to the track and not slip). If initially the wheelset is centered on the railroad track then the effective diameters of each wheel are the same and the wheelset rolls down the track in a perfectly straight line forever. But if the wheelset is a little off center so that the effective diameters (or radii) are different, then the wheelset starts to move in a curve of radius (depending on these wheelset radii, etc. ; to be derived later on). The problem is to use kinematic reasoning to find the trajectory of the wheelset, or more precisely, the trajectory of the center of the wheelset projected vertically on the roadbed in the center of the track. This is a trajectory on the plane of the level earth's surface and plotted on an graphical plot where is the distance along the railroad and is the "tracking error", the deviation of the center of the wheelset from the straight line of the railway running down the center of the track (midway between the two rails). To illustrate that a wheelset trajectory follows a curved path, one may place a nail or screw on a flat table top and give it a push. It will roll in a circular curve because the nail or screw is like a wheelset with extremely different diameter wheels. The head is analogous to a large diameter wheel and the pointed end is like a small diameter wheel. While the nail or screw will turn around in a full circle (and more) the railroad wheelset behaves differently because as soon at it starts to turn in a curve, the effective diameters change in such a way as to decrease the curvature of the path. Note that "radius" and "curvature" refer to the curvature of the trajectory of the wheelset and not the curvature of the railway since this is perfectly straight track. As the wheelset rolls on, the curvature decreases until the wheels reach the point where their effective diameters are equal and the path is no longer curving. But the trajectory has a slope at this point (it is a straight line which crosses diagonally over the centerline of the track) so that it overshoots the centerline of the track and the effective diameters reverse (the formerly smaller diameter wheel becomes the larger diameter and conversely). This results in the wheelset moving in a curve in the opposite direction. Again it overshoots the centerline and this phenomenon continues indefinitely with the wheelset oscillating from side to side. Note that the wheel flange never makes contact with the rail. In this model, the rails are assumed to always contact the wheel tread along the same line on the rail head which assumes that the rails are knife edge and only make contact with the wheel tread along a line (of zero width).

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. Conventionally a railway vehicle has stable motion in low speeds, when it reaches to high speeds stability changes to unstable form. The main purpose of nonlinear analysis of rail vehicle system dynamics is to show the view of analytical investigation of bifurcation, nonlinear lateral stability and hunting behavior of rail vehicles in a tangent track. This study describes the Bogoliubov method for the analysis. Two main matters, namely assuming the body as a fixed support and influence of the nonlinear elements in calculation of the hunting speed, are mostly focused in studies. A real railway vehicle has many more degrees of freedom and, consequently, may have more than one critical speed; 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.

Many road–rail vehicles feature independent axles and suspension systems on each rail wheel. When this is combined with the presence of road wheels on the rail it becomes difficult to use the formulae above. Historically, road–rail vehicles have their front wheels set slightly toe in, which has been found to minimise hunting whilst the vehicle is being driven on rail.