Why trains stay on tracks

In this article, we give both qualitative and quantitative explanations of why a train stays on its track, in spite of perturbations that could cause it to derail. We show that train stability originates from the conical shape of the wheels, which gives rise to a restoring normal force in response to a lateral disturbance. We first demonstrate translational stabilization in a simple situation where the rails are assumed frictionless and the steering motion of the wheel is neglected. We then develop a more comprehensive model, taking friction and steering into account. We show that rolling friction couples the rotational motion to the translational motion, enhancing overall stability. Finally, we find approximate formulae for the parameters governing stability, and show good agreement with parameters of a real railway coach.