The model of the bicycle is described in Fig. 1. L is the distance between the contact points of the front and back wheel respectively; A is the distance between the back wheel contact point and S, the projection of the center of mass. Typical values of the parameters are L = 1.05 m, A = 0.33 m, H = 1.2 m, r = 0.36 m, Φ = 70 deg. The mass of rider and frame M is taken as 80 kg. Unless stated otherwise these parameter will be used in all calculations.

Fig.1

Key bike geometry parameters. The origin of the coordinate system is in the steering axis and at the center of rotation of the wheel around the steering axis. σ is the turn angle of the handle bar around the steering axis, θ_f is the lean angle of the frame and θ_cm is the lean angle of the center of mass.

Fig. 2

Kinematics of a bicycle in a turn.

α is the angle between the instantaneous direction of the front wheel and the x-axis, α_cm  the corresponding angle of the instantaneous direction of S. R_f, R_cm and R_r are the radii of curvature of front wheel, center of mass projection S and rear wheel respectively.

Note: Unless the steering axis is vertical, the angle α of the front wheel trajectory is different from the turn angle σ of the handle bar. For small angles the relation can be approximated as

The general expression is:

It is easily found that

For α # 0 the velocity varies along the x-axis. In all forthcoming expressions, v is defined as the velocity of the point S. Also the radius of curvature is different for each point along the x-axis. For instance for α = 90° the radius of curvature is proportional the distance from the rear wheel. The radius of curvature for point S is given by:

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