using the speed of the vehicle , which is assumed constant: The angular acceleration of the axle in yaw is: The inertial moment (ignoring gyroscopic effects) is: where is the force acting along the rails and is the moment of inertia of the wheel set. the maximum frictional force between the wheel and rail is given by: where is the axle load and is the coefficient of friction. Gross slipping will occur at a combination of speed and axle deflection given by: 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 (creep slippage). During normal operation these forces are well within the limiting friction constraint. A complete analysis takes these forces into account, using rolling contact mechanics theories. However, the kinematic analysis assumed that there was no slippage at all at the wheel-rail contact. Now it is clear that there is some creep slippage which makes the calculated sinusoidal trajectory of the wheelset (per Klingel's formula) not exactly correct. 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: Using the operator: the angular acceleration equation may be expressed in terms of the angular velocity in yaw, : integrating: so the kinetic energy due to rotation is: 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: (to second order of small quantities). the displacement of the support point out from the centres of the treads is: the axle load falls by The work done by lowering the axle load is therefore: This is energy lost from the system, so in order for the motion to continue, an equal amount of