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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 Tlean_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 θcm the lean angle of the center of gravity. Tlean_g tends to overturn the bicycle and has to be compensated by a second and opposite torque Tlean_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, Rcm the radius of curvature of the center of mass and αcm 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 αcm 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 Tlean_gyr/Tlean_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 dαcm/dt term in Tlean_c . Its contribution is also negligible, because of similar reasons.


Figure 3
In a stationary equilibrium the vector sum of the centrifugal force and gravity falls into the axis center of mass - ground contact.


The equilibrium condition for the lean angle θcm becomes


or

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.

From the above equation it becomes clear that there are two velocity regimes. At high velocity, equilibrium for a given lean angle is achieved by the centrifugal force (first term in parenthesis). Below a transition velocity vt, the centrifugal force is no longer able to maintain equilibrium. Because the second term, the kink force, varies only with v and not with v2, it is able to maintain a dynamic equilibrium at smaller velocities. The transition between the two regimes is easily felt when riding a bike. Somewhat below pedestrian velocity, equilibrium is maintained by a succession of rapid turns on the handle bar. At higher speed it is the position of the handle bar and not its variation which controls the lean angle.

The transition velocity, i.e. the velocity at which the two terms in parenthesis become equal is given by


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

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.

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