Stable and Unstable Systems

Abstract

Most practical systems are required to be stable, at least dynamically. Consider a bicycle; it is statically unstable, yet a small child can learn to ride it. How is intrinsic stability obtained? There is a popular notion that the head angle confers stability, yet early bicycles had upright steering heads. A little thought shows that sloping heads actually tend to de-stabilise. Under gravity the load and earth try to come together. Relative to the bicycle, the ground point P as defined in figure 26 tries to come upwards from the lowest position at straight ahead, which it does by rotating the steering. This is easily confirmed in practice; the equilibrium point comes at a steering angle of 60 to 80° from straight. The stability comes from the trail. If the bicycle leans sideways, the ground-reaction has a lateral component steering the wheel towards the leaning side until centrifugal force restores balance. If the trail accidentally becomes negative, riding hands-off becomes very difficult. Some small-wheeled bicycles were designed on the basis of head angle and swept-forward forks, leaving too little trail.