 Figure 4 Geometry of front wheel and fork. k is the fork offset or rake and Δ the trail. In mountain bikes k is realized by a parallel offset or by a kink in the fork. Typically the angle Φ of the steering axis is around 700 . This results in a trail, which for k = 0 would be around 12 cm, but which is reduced to around 6 cm by a rake of 6 cm. 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 θf the lean angle of the frame. Furthermore:  p parametrizes the wheel circumference. The origin of p is at the intersection with the x1 axis. pmin is the coordinate of the contact point K (Figure 4). Readers not interested in the derivation of pmin 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  To compute pmin, U(p,σ) has to be first transformed into the conventional x,y,z coordinate system. pmin is the value of p at which the first derivative with respect to p of the z-component of U is zero. This is the angle of minimum z-coordinate, i.e. the coordinate of the contact point K. Utransf is U expressed in standard x,y,z coordinates.  with a z-component of:  The first derivative with respect to p becomes  with a zero value at  This results in   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 (θf = 0) with σ = 0, the trail becomes  The stable equilibrium handle bar turn angle for a bicycle at rest is given by Δscalar = 0. For an upright bicycle (θf = 0) a surprisingly simple expression for Δscalar = 0 results:  For larger values of σ, Δscalar becomes positive. σzero thus defines the stability limit for an upright bicycle. For a standard bicycle geometry, σzero is slightly larger than 600.  Figure 5 Polar plot of Δscalar (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 Δscalar is negative (small loop), for all other angles it is positive (large loop).  Figure 6 Dependence of the trail on lean angle. The plus sign of the lean angle corresponds to a lean in the direction of the handle bar turn angle. As shown in Figure 6, the turn angle at which the trail becomes zero rapidly decreases when the bicycle is tilted in the turn direction. |