Nearly everybody knows how to ride a bicycle. Nevertheless the knowledge of the physical principles which are relevant for bicycle stability is not widespread and obscured by persisting myth and folklore. Early models of bicycle stability were based on gyroscopic torques. The mass of the wheels is much smaller than the total mass of bicycle and rider. Also the center of mass of the system is about a factor of three higher than the hub, the center of mass of the wheels. This means that the gyroscopic torque is negligible compared to the torque due to the centrifugal force. Since both scale with velocity v squared, this is true at any speed.

Probably the most important contribution to the understanding of bicycle physics is due to David Jones (Physics Today 23, April 1970) [1]. His first attempt to design an unridable bicycle by eliminating gyroscopic torques failed. Bicycles with tiny ball bearings instead of wheels proved to be perfectly ridable. Also compensating or even overcompensating gyroscopic torques by an additional dummy wheel turning in the opposite direction had no effect on rideability. His second attempt was successful. Bicycles which had a negative trail, i.e. a contact point K in front of the projection of the steering axis (see Fig. 1) were unridable. This demonstrated the importance of the steering geometry for bicycle riding.

For a bicycle at stationary turn radius and not very low speed it is the torque due to the **centrifugal force** which compensates the lean torque. In equilibrium the sum vector of centrifugal force and gravitational force lies in the plane center of mass – ground contact line of the wheels. Since the centrifugal force scales with v^{2} (v = velocity), it becomes too weak to compensate the lean torque at very small speed. It is then substituted by a related force, the **kink force**. If the handle bar turn angle σ is instantly changed, then the trajectory of the front wheel exhibits a kink. A change at finite speed dσ/dt induces a transient small radius of curvature instead of a kink. This results in a force proportional to v dσ/dt. The kink force thus decreases slower in speed compared to the centrifugal force and dominates at very low speed. The transition from centrifugal force regime to the kink force regime is easily noticeable. It manifests itself by the onset of rapid corrective handle bar movements at very small speed. At high speed the kink force is used to initiate a turn. A rapid handle bar movement in the “wrong” direction induces a kink force which rapidly tilts the center of gravity into the turn.

More complex and crucial for hands-free riding is the **equilibrium of the torques acting on the handle bar**. Jones was the first to show that the **steering geometry** and thus the torques due to forces acting on the contact point K (Fig. 1) of the front wheel are dominating. The trail Δ acts as lever arm for the upward normal force and the inward centripetal force. The torque due to the normal force tends to turn the handle bar outwards, whereas the torque of the centripetal force tends to turn the handle bar inwards. The stationary gyroscopic torque of the front wheel is parallel to the lean axis and has a component in the steering axis if Φ is different from 90°. Since the component scales with v^{2} it can be included as a correction factor F to the centrifugal torque. For typical mass- and geometry conditions, F equals approximately 1.12. For small lean and turn angles the equilibrium turn angle of the handle bar is given by

where θ_{r} is the lean angle of the frame (not of the center of mass!), L the distance of the ground contact points of the wheels, and g the gravitational acceleration. Obviously the equation has only a solution above the velocity v_{crit}

The key parameter controlling v_{crit} is the steering axis angle Φ. A vertical steering axis would result in v_{crit} = 0. Typical values of Φ are in the range of 70° which results in v_{crit} = 6 – 7 km/h.

Equilibrium against tilting at a given center of mass lean angle θ_{s} is given by

where α is the angle between the front wheel trajectory and the horizontal axis of the frame. α is different from σ, the turn angle of the handle bar. For small angles the relation is given by α ≈ sin (φ) σ.

Hands-free riding requires that the equilibrium turn angle σ_{0} of the handle bar equals the angle required to provide equilibrium of the center of mass at a given lean angle θ_{s}. This can be achieved by an appropriate lean angle of the frame θ_{r} at a given lean angle θ_{s} of the center of mass. In equilibrium and for small angles the ratio of the two lean angles is given by

where F ≈ 1.12 is a correction factor due to the gyroscopic torque of the front wheel. The only control parameter available for the rider under hands-free conditions is V. It can be used for steering into turns or to provide stability. Fig. 2 shows that above v = 18 km/h the frame has to be tilted slightly more than the center of mass. With decreasing speed, V becomes smaller and smaller. At v_{crit} even an infinitesimal angle θ_{r} results in an uncontrollable handle bar turn angle.

For normal hands-on riding, the control mechanism illustrated above is nice to have, but not mandatory. The handle bar equilibrium can be offset to a useless position, by attaching a large weight to one side of the handle bar. Nevertheless riding is still possible, albeit slightly less convenient.

Below v_{crit} riding with hands on the handle bar is no problem. It is thus of interest to discuss the **handle bar equilibrium at v < v _{crit}**. For the bicycle at v = 0 only the torque due to the normal force is acting on the handle bar. The equation torque = 0 has two types of solutions. The first is given by the turn angle σ at which the torque has no component in the steering axis. For the upright bicycle this is fulfilled at σ = 0. This solution is not stable at v = 0. The stable solution is given by the condition lever arm = trail = zero and thus torque = 0. For the upright bicycle (θ

The hub offset k (see Fig.1) is the key parameter for the equilibrium of the handle bar at v = 0. For k = 0, the equilibrium angle would be 90°. A bicycle pushed at the saddle with a free handle bar would stall. An offset k of 6 cm at r = 36 cm and Φ = 70° reduces σ_{zero} to about 60°. From an ease of handling point of view, a small σ_{zero} is preferable. On the other hand σ_{zero} reflects the limit of the stability range. For σ > σ_{zero} the trail has the opposite sign. σ_{zero} ≈ 60° is thus a compromise between stability range and ease of handling.

We now discuss the velocity range 0 < v < v_{crit}. The turn angle equilibria are shown in Fig. 3. For v = 0 the trail = 0 solution σ_{zero} decreases with increasing lean angle. Solutions with a finite lever arm (trail different from zero) are only possible above about 4 km/h. However, even at zero lean angle the symmetric σ_{zero} = 0 solution is not stable below v_{crit}. A reasonable quasilinear range, required for hands-free riding, is only found above about 10 km/h. The quasilinear range is limited by the trail = 0 solution, which forms the boundary of stability.

Fig. 3 also shows that high quality bicycles have reached an optimized geometry. In the centrifugal force controlled regime, a quasilinear σ_{0}(θ_{r}) is desired. This allows hands-free riding. On the other hand, at very low speed, in the kink force controlled regime, a stable small angle solution for the handle bar is not desirable. The instability of the small angle solution supports the corrective turns on the handle bar. The optimum v_{crit} is thus at the transition between the two regimes. A steering axis angle around 70° provides a v_{crit} of 6 – 7 km/h and the full instability below 4 km/h (Fig. 3). Also a hub offset k of 6 cm at r = 36 cm and Φ = 70° gives a good compromise between low speed handling and stability range.

A discussion of the dynamic properties is beyond the scope of this short contribution. The linearised differential equation for lean angle and handle bar turn angle respectively are coupled and of second order. For a rigid system (θ_{r} = θ_{s}) at hands-free riding, there are two equations and two unknowns (θ_{s} and σ). From Fig. 3 it is evident, that above about 18 km/h a rigid system will suffer a capsize instability. Below about 15 km/h exponentially growing oscillations prevent stability. In reality the mobile hip of the rider decouples θ_{r} and θ_{s} and provides an efficient suppression of the oscillations. Gyroscopic torques due to dσ/dt have no component in the steering axis. However, at high speed gyroscopic torques proportional to dθ_{r}/dt contribute significantly to the total handle bar torque at special riding conditions such as counter steering and hip-thrust to initiate a turn. Qualitatively the gyroscopic dθ_{r}/dt term acts as a negative damping.

A more detailed discussion of bicycle physics can be found at http://sites.google.com/site/bikephysics/Home

*Table 1: Summary of the influence of geometry parameters on riding properties.*

Geometry parameter | Influence on riding properties |

Steering axis angle Φ | - Generates trail - Defines v _{crit}- Has to be chosen such, that v _{crit} is at the lower end of the centrifugal force regime. This leads to Φ ≈ 70°. |

Trail Δ | - Leads to quasilinear handle bar equilibrium above v_{crit} which allows hands-free riding- Leads to instability of small angle solutions of the handle bar equilibrium below v _{crit}. This supports the corrective motions in the kink force regime. |

Hub offset k | - Reduces the equilibrium turn angle of the handle bar at v ≈ 0, which makes handling easier - Reduces the stability range (range with trail > 0) - Reduces the tail and thus the amplification factor of the control system - Optimum value given by compromise between ease of handling at v ≈ 0 and stability range. |

[1] http://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/vol59no9p51_56.pdf

Hansruedi Zeller has a PhD in physics from the ETH Zürich. He held positions in industrial research at General Electric USA and BBC/ABB in Switzerland. He was Vice President Technology at ABB Semiconductors and before his retirement Senior Consultant at Consenec AG.

*[Released: July 2009]*