Here a few remarks of my former mathematics student Sven Wallbaum. He describes the topic comprehensible and funny at the same time. It is about forces that affect us while riding a motorcycle and how they can be described mathematically. Voila:
What keeps my motorcycle on the ground?
Adhesion… an essential point with motorized two-wheelers. To be more precise: the holding force caused by the friction between the tyre and the road surface.
The idea behind it is as simple as it is decisive: As soon as the maximum holding force is exceeded, it becomes tricky.
How do I know how far I can drive my motorcycle?
The maximum holding force depends linearly on the coefficient of friction, which results as a material constant from the combination of tyre material and road surface, and the normal force, the force which acts perpendicularly on the road surface: In general, it is mainly the weight force of the motorcycle and the rider that is added, with an inclined road surface the influences of the centripetal force etc. are proportional.
The normal force is relevant for static friction, it is perpendicular to the “road surface” and thus also perpendicular to the static force. It is the force with which the two materials rubber and asphalt (depending on the material combination µ) are pressed against each other.
The transverse force is parallel to the road surface and therefore does not influence the adhesive force. It is important, however, that the adhesive force is greater than the shear force, otherwise, you will take off. The part that the shear force is smaller than the holding force can then be put into the acceleration force circle.
What role does the tangential force play in motorcycling?
And why I should know about it!
The second interesting force is the tangential force, which is generally composed of the centripetal force and the acceleration force.
Circle shows the ratio of lateral guidance forces to acceleration or braking forces in inclined positions.
Straight ahead with the bike
It is obvious that when driving straight ahead, the grip is lost (i.e. wheelspin or slipping wheel) as soon as the acceleration or braking force exceeds the maximum holding force.
The same applies to the corner: the grip is lost as soon as the centripetal force exceeds the maximum holding force.
But what about the combination of both forces?
The answer is relatively simple. Since the two forces are orthogonal to each other (acceleration in the direction of travel, centripetal force transverse to the direction of travel), the resulting force can easily be determined using the Pythagorean theorem.
The resulting force is of interest because the departure threatens as soon as it exceeds the maximum holding force. It does not matter in which direction the resulting force acts. From this consideration – and the two-dimensionality of the road surface – results in Kamms’ circle, a circle whose radius represents the maximum holding force at a given point in time. As long as the resulting force (vector starting from the centre of the circle, the length of which represents the amount of force) is within the circle, traction is ensured.
Why is Kamms’ circle interesting at all?
The answer is simple: It visualizes the non-linear relationship between the forces!
A short excursion into mechanical mathematics
Let F_h_max be the maximum holding force, F_b_max = F_h_max the maximum acceleration/braking force and F_z_max = F_h_max the maximum centripetal force. With the help of Kamms’ circle you can now see that at 90% of the maximum centripetal force, for example, you can still accelerate with about 43.5% of the maximum acceleration force, since
sqrt(0.9²*0.435²) approx 1.