3. Torques in the Lean Axis - Stability against Capsizing

In this section we discuss the torques which act on the lean axis (see Figs. 1 and 2 in section "Geometry and Kinematics"). Equilibrium against capsizing is given by the condition that the sum of all torques is zero.

A first torque T_{lean_g} results from gravity acting on bicycle and rider. This torque is given by:

H is the height of the center of mass, M the total mass of the system, g the gravitational acceleration and θ_cm the lean angle of the center of gravity. T_{lean_g} tends to overturn the bicycle and has to be compensated by a second and opposite torque T_{lean_c} which results from a generalized centrifugal force. This torque is given by

where a is the acceleration of point S in the y- direction (see Fig. 1 in section "Geometry and Kinematics"). The acceleration a is given by

with v = velocity of the center of mass, R_cm the radius of curvature of the center of mass and α_cm the angle between the trajectory of the center of mass and the frame axis x. The first term in the bracket relates to the conventional centrifugal force, the second term is special to a bicycle. If the handle bar of a bicycle is abruptly turned, then the trajectory of the front wheel exhibits a kink. The rear wheel on the other hand has a smooth trajectory. The center of mass experiences a reduced kink. For a finite turning speed of the handle bar, the kink is replaced by a reduced radius of curvature. This reduced radius of curvature gives raise to a force related to the centrifugal force, the kink force.

There is also a gyroscopic torque exerted from the rotating wheels on the fork and on the hub of the back wheel respectively. Because this torque is negligible, only the first order term valid for small values of α_cm is given:

I₀ is the moment of inertia of the wheel, r the radius of the wheel and A the horizontal distance between the rear hub and the center of mass (see Fig. 1). This torque also counteracts the lean torque resulting from gravity.

If I₀ is approximated by mr² where m is the mass of rim and tire of the wheel, then the ratio T_{lean_gyr}/T_{lean_c} for a stationary turn becomes to first order:

The ratio r/H is approx. 1/3 and the ratio m/M is about 1/100. It is important to note that the ratio of the two torques is independent of velocity. The conclusion thus is, that the gyroscopic torque is negligible compared to the torque of the centrifugal force. For a toy bicycle without a rider, in which the mass is centered in the wheels, the ratio is close to unity and the gyroscopic torque is important. Such bicycles are often used in physics lectures and mislead the students about the true nature of bicycle stability.

There is also a gyroscopic torque proportional to the product v dσ/dt (σ = turn angle of the handle bar). This torque has to be compared to the dα_cm/dt term in T_{lean_c} . Its contribution is also negligible, because of similar reasons.

which for small angles can be linearized as

or in a stationary equilibrium

where Φ is the angle of the steering axis (Fig. 1) and σ the turn angle of the handle bar.

In a nonstationary situation the second term in the equation for θ_cm has to be included. Because the second term, the kink force, varies only with v and not with v², its importance increases with decreasing velocity. The transition velocity, i.e. the velocity at which the two terms in parenthesis become equal is given by

where t_t is a characteristic time for a correction on the handle bar. With A = 33 cm and t_t = 0.2 sec, v_t becomes 1.65 m/sec or about 6 km/h.

At small velocities continuous corrective motions of the handle bar are required. If the bicycle suffers at t = 0 a step perturbation pert of the form pert = const θ_cm/dt then the rider will counteract by a turn of the handle bar. In a first phase the bar is turned out, in the second phase it is turned back. The kink force starts immediately with the turn of the handle bar. The action of the centrifugal force sets in later. It is proportional to the turn angle. In the second phase the kink force changes sign and thus reduces the sum force.

Note that the kink force is proportional to A. The stabilizing effect of the kink force can be increased by increasing A, i.e. by shifting the center of mass towards the handle bar. Mountain bikers know that when riding an extremely steep uphill at very small speed, balancing the bicycle becomes very difficult. Due to the slope, the center of mass shifts to the back, A decreases and even may tend to zero. This happens when the front wheel is close to loosing contact with ground. The countermeasure is "biting the handle bar" to increase A.

The kink force has a simple physical interpretation. The effect of the corrective motions at the handle bar is, to continuously correct the position of S, such that it falls below the center of mass. In other words, the centrifugal force brings the center of mass above S and the kink force brings S below the center of mass.

Since the kink force is linear in v, it reverses sign if the velocity is reversed. This is opposed to the centrifugal force which scales as v² and thus does not change sign. Backward riding is equivalent to forward riding with a rear wheel steered bicycle. In such a bicycle the kink force does not damp, but promote oscillations. Rear wheel steered bicycles are thus expected to be unridable. This has been confirmed by experiment.

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