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How come the wheel centre O shifts to O* instead of O₁ in the case where we have a non-zero fork offset, when turned sideways? Shouldn't it have shifted to O₁, such that the contact point P₁ lies exactly below the centre, as is the case of zero-fork offset? Or maybe, P₁ could've lied directly below O*, if what the author says is true? Can't really wrap my head around the idea. Could anyone explain? These are the assumptions that the author made:

  • the roll angle of the motorcycle is zero.
  • the wheels have zero thickness.

(Book : Motorcycle Dynamics by Vittore Cossalter) enter image description here

enter image description here

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  • $\begingroup$ Shouldn't it have shifted to O₁, such that the contact point P₁ lies exactly below the center, as is the case of zero-fork offset? No. For nonzero axis angles, the contact point only lies below the wheel center if the steering angle is zero. When you have steering and a nonzero axis, it gets quite complicated. But the offset doesn't change the relationship between the center and the contact point. $\endgroup$
    – Phil Sweet
    Jan 6 at 14:38
  • $\begingroup$ And how did you render the subscript such that it copied into comments. I've never been able to do that. $\endgroup$
    – Phil Sweet
    Jan 6 at 14:40

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Both images hold the steering axis rigid in space and scrub the tire across the floor as the steering is changed. Normally the tire rolls on the floor and the frame moves in space.

With the axis fixed, the when the bike lowers, it must also slide forward along the axis. So we need to be conscious of terminology and what directions things are constrained to move in. If you change the rules so that the bike's back wheel is stationary and the front wheel rolls and doesn't slip as you turn the steering, the results look very different, but it is a lot harder to draw the picture. But we don't really care about what happens when the bike is stopped. This is about getting the information to predict behavior at speed. And the difference between the two viewpionts (the difference in the trajectories) is negligible at speed. So we use a somewhat odd axis-centric coordinate system to get the important geometrical parameters. So this is what you see in the sketches - the intersection of the steering axis and the ground is fixed.

In the upper sketch, the wheel center is swung to the right when steered right. The bike is then lowered along the steering axis until the tire touches. Point 0* is after the steering input, and point O₁ is 0* slid down the axis. It is the new wheel center.

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