Bike Steering Geometry Details

Some aspects of the unusual geometry of this bike are further dealt with here.


In order to avoid confusion, the following diagram illustrates the terminology used here. It shows in side view one headset (in red) that has a positive steering axis angle (approx. 15 degrees away from the vertical), and a second headset (in green) that has a negative steering axis angle (approx. 7 degrees). The steering axis is the line passing through the center of the headset, projected to the ground. A vertical line projected down from the wheel axle intersects the ground at the center of the contact patch between tyre and road surface. The distance between this point and the point of intersection between steering axis and road surface is the ground trail, measured on the road surface.

Click on the diagram for a larger view.

In this diagram the trail in both cases is positive, because the contact patch lies behind the steering axis irrespective of its angle. The perpendicular distance between contact patch and steering axis is the effective trail. Ground trail and effective trail differ in that the latter is the actual length of the lever arm on which the side force acts when there is a slip angle. Slip angle occurs when the steering is turned and the contact patch is displaced away from the line of travel (as seen from above). This displacement causes a torque around the steering axis which tends to restore the steering to the straight-ahead position if the trail is positive. This effect is also referred to as the castor effect, which is the self-centering observed in castor wheels.

Other terms for steering axis angle are castor angle (or caster in American English) and rake angle. There is a close relation between steering axis angle and trail that is mediated by the fork offset which is the perpendicular distance between the wheel axle (or spindle) and the steering axis. For a given steering axis angle the trail depends on the offset. Positive offset is when the wheel spindle is situated in front of the steering axis. Negative offset is when the wheel spindle is situated behind the steering axis.

In the diagram we see a small positive offset is required to obtain the given trail with a 15 degree steering axis angle. The diagram clearly shows that with a suitable offset a headset with either postive or negative angle can have equal amounts of positive trail, but when the steering axis angle is negative the offset must be negative and quite large to obtain the same amount of positive trail.

Comparisons in terms of handling

Bicycle designs with a negative steering axis angle are quite unusual and for this reason there is very little information on their handling performance. It appears, however, that a small negative steering axis angle (less than 10 degress) may be superior to positive angles. From personal experience, a negative angle is not harmful to handling, and no theoretical argument that this author is aware of contradicts that experience.

Another exemplar is Frankenbike by Jeff Del Papa.

The following diagram illustrates two points which suggest that a negative angle is superior to a positive angle in terms of handling. Again it is a side-view using the same colour coding (red = positive, green = negative), but here the negative angle (15 degrees) has been exaggerated in order to show the difference more clearly. The diagram is a scale drawing of the lower part of a 20" wheel that has run up against a 2 cm obstruction in the road. The ground trail is 5 cm.

Click on the diagram for a larger view.

Negative castor

When a front wheel encounters an obstruction a situation of negative castor occurs, which is an unstable equilibrium. As shown in the diagram, the contact point with the obstruction lies ahead of the steering axis, and consequently the effect of trail is reversed. If the obstruction is not perfectly in line with the direction of travel then the result will be disequilibrium and a disturbance in the direction of travel. The diagram shows, however, that the disturbing force of the obstruction will be smaller in the case of a negative steering axis angle than when that angle is positive. This is because the length of the lever arm through which the obstructing force acts is shorter when the steering axis is inclined forwards.

In the diagram the solid red and green lines perpendicular to the steering axis show the relative lengths of the lever arm. The difference here is exaggerated, because a 15 degrees negative angle is rarely seen in practice. More realistically, with a 7 degrees negative angle compared to a 15 degrees positive angle the difference in leverage is approximately 10%. Such a difference could represent a noticeable handling superiority in favour of negative steering axis angles.

Steering head drop

The dotted red and green lines drawn perpendicular to the respective steering axes, indicate the effective trail in both cases. Note that for negative steering axis angle the effective trail arm lies below the road surface. This fact implies that with the bike held vertically, as the steering is turned to either side away from straight forward, the steering head will rise. Consequently the steering head is in its lowest position with respect to the ground when steering in a straight line. Because gravity will assist in keeping it that way, this is a stable equilibrium.

In the case of positive steering axis angle the opposite applies: The steering head will fall whichever way the steering is turned, and we have an unstable equilibrium.

While these two equilibria clearly differ, their actual effect on handling may be very small or not apparent. However, the effect of a negative angle, if noticeable, would seem to be the more desireable of the two.

Other differences

The amount of fork offset has implications for handling because it seems to determine the likelihood of steering wobble.

Any deflection of the wheel by an external force is countered by torque resulting from the slip angle, as already explained. Gyroscopic reactions can also deflect the wheel about the steering axis. Offset moves the center of gravity of the wheel away from the steering axis and thus increases the moment it has about the axis. The torque due to slip angle may be insufficient to counter these reactions and the result could be an uncontrolled wobble. Zero offset would minimize this likelihood and maximize the self-steering effect of positive trail.

As stated above, to obtain positive trail a negative angle demands a relatively large offset, while the same trail can be achieved with zero offset by a small positive steering axis angle. The difference may be very small if the mass of the wheel is small, but in the case of a front-wheel drive that mass can be quite large, and its interaction with other forces is compounded by the offset. This clearly is a disadvantage on the side of negative angles.

Conclusion about handling

Given the sparse data and the small magnitude of demonstrable theoretical advantages, the most appropriate conclusion may be that it does not really matter if your steering axis angle is positive or negative. Other factors such as drivetrain, frame construction and appearance are equally important in choosing a particular geometry.

Conclusion about construction

It has already been mentioned that the forward location of a headset with negative angle is advantageous for the construction of front-wheel drive designs. One disadvantage of the negative angle in any design is seen in the diagram above. When the front wheel encounters a bump there will be more sheer stress in the negatively inclined fork. The direction of force applied is almost at right angles to the negatively inclined steering axis, and more along the line of the positively inclined steering axis. This would mean that a stronger, heavier fork is required. Presumably, a BMX fork designed for jumping would be adequate.
Updated: 15 June 2000