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According to vehicle dynamics, there appears to be two different ways a wheel travels a curved path. I don't think this requires an expert in vehicle dynamics because it is a simple physics problem of a rolling wheel changing direction.

The figures below examine the path of the front wheel of a bicycle as the rider counter-steers and proceeds through a left turn - Figs 1A-1D. enter image description here As the bike rolls, there are periods when the rider is turning the front wheel and periods when the rider holds the front wheel at a fixed angle - Figs 2A-D. During the “wheel turning” periods, which are shown in red, the front wheel follows a curved path. Here, the wheel changes direction by rotating/turning about its contact point. Pretty simple stuff because it’s easy to understand how turning a rolling wheel results in curved path travel.

enter image description here

During the periods where the front wheel is fixed at an angle, which are shown in blue, the wheel also proceeds through a curved path. We now say the wheel proceeds through the curved path by a different mechanical principle and no longer turns at its contact point to change travel direction. Since the wheel directions are fixed, we think they turn about a location other than at each contact point. We believe both wheels are tangential velocities rotating about the turn center - blue paths in Fig 3.

enter image description here

How can the front wheel change travel direction by rotating about its contact point (red paths) and then mysteriously change the way it changes its travel direction (blue paths)?

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2 Answers 2

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The turning of a front bicycle wheel is about an axis determined by the axle which mounts the handle bars. On an automobile, the turning axle is beside a steerable wheel and out of sight behind the wheel when viewed from outside.

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  • $\begingroup$ And when you’re not turning it, how is it turning? $\endgroup$
    – Matt Zusy
    Dec 16, 2021 at 17:07
  • $\begingroup$ If you are not holding the handle bars and lean, the wheel will precess about that same axis. $\endgroup$
    – R.W. Bird
    Dec 17, 2021 at 14:35
  • $\begingroup$ So the wheel changes the way it changes direction at the instant you stop turning the handle bars. How does it know that it should to do that? What changes mechanically at that instant? $\endgroup$
    – Matt Zusy
    Dec 17, 2021 at 17:23
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If you consider the center of the axle, it has forwards motion and rotation at the same time. The combined effect is that the center of rotation (pivot point) is a the distance point indicated in your diagrams.

fig

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  • $\begingroup$ Actually I think an axle only allows linear motion and not rotation unless the axle itself is rotated, which is what happens while we turn the front wheel of the bike. When we hold the axle at an angle as in your image, does the wheel turn (change direction) without the axle turning (rotate on veticalish axis)? $\endgroup$
    – Matt Zusy
    Dec 16, 2021 at 17:51
  • $\begingroup$ @MattZusy from an outside observer it does change direction, as the whole bike does. In fact, it pivots about the point shown above, and each velocity vector (also shown) can be derived from the rotation about the pivot. $\endgroup$
    – JAlex
    Dec 16, 2021 at 18:57
  • $\begingroup$ Thanks @JAlex. Think about the front wheel alone. When it’s rolling, what would you have to do to make it turn and create a curved path? You have to rotate it on a vertical axis, through its contact patch, while it rolls. Put the front wheel back on the bike, fix it at an angle and roll the bike. It now rolls along a curved path. How? Because it follows a rule or because it rotates on a vertical axis through its contact patch? $\endgroup$
    – Matt Zusy
    Dec 16, 2021 at 19:36

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