4. The Trail

As Jones has first pointed out and experimentally demonstrated, the trail plays a key role in bicycle stability. Because of the trail, the forces acting on the ground contact point K exert a torque on the steering axis. This torque is under normal riding conditions the dominating torque and much larger than gyroscopic torques acting on the steering axis.The computation of the trail as a function of lean angle of the frame and of the turn angle of the handle bar is lengthy, but results in a reasonably simple final result. To compute the trail, we use a different coordinate system, which makes full use of the symmetry of the problem. x₁ is in the plane of the frame and perpendicular to the steering axis and z₁ is in the steering axis. In this coordinate system the trail vector has the form:where σ is the turn angle of the handle bar and θ_f the lean angle of the frame. Furthermore:

p parametrizes the wheel circumference. The origin of p is at the intersection with the x₁ axis. p_min is the coordinate of the contact point K (Figure 5).

Readers not interested in the derivation of p_min can ignore the following paragraph in italics.

For a handle bar angle σ, the circumference coordinate U (in the coordinate system of Fig.4) in terms of the angle p is given by

This results in

with

The sign convention in the x₁ coordinate system is such, that a positive trail corresponds to a negative value of Δ.

For the upright bicycle (θ_f = 0) with σ = 0, the trail becomes

There is a confusion about the role of the trail for bicycle stability. Stable no-hands riding requires that the upright and symmetrical bicycle position (θ_f = σ = 0) corresponds to a maximum in potential energy. The gravity induced torque on the handle bar will then be such that the handle bar turns into the lean for θ_f different from zero. Otherwise no negative feedback to counteract the capsize instability is at work. A maximum in potential energy is achieved if Δ_scalar (0,0) is negative. A maximum can be obtained also by other means, e.g. by a very weird mass distribution of handle bar and frame (1). This has nothing to do with the stability of a standard bicycle. In the next paragraph it is shown that the trail can change sign as a function of lean and turn angle. In particular Δ_scalar(0,σ) is positive for σ = 180⁰ if Δ_scalar(0,0) is negative. This, however, is an artefact of the coordinate system. A bicycle with the handle bar turned by 180⁰ is perfectly ridable. This is because also this position exhibits a maximum in potential energy.

The stable equilibrium handle bar turn angle for a bicycle at rest is given by Δ_scalar = 0. For an upright bicycle (θ_f = 0) a surprisingly simple expression for Δ_scalar = 0 results:

For larger values of σ, Δ_scalar becomes positive. σ_zero thus defines the stability limit for an upright bicycle with respect to the σ = 0 equilibrium position. After the sign change the reference position for stability is now the σ = 180⁰ equilibrium position. For a standard bicycle geometry, σ_zero is slightly larger than 60⁰.

Figure 6Polar plot of Δ_scalar (negative trail) as a function of the handle bar turn angle for an upright bicycle with steering axis angle Φ = 70⁰ and hub offset k = 6 cm. For approximately -60⁰< σ < 60⁰ Δ_scalar is negative (small loop), for all other angles it is positive (large loop).