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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)?

In the first image below: A car corners with the wheels leaving behind concentric circular travel paths and we then find a point we call the turn center. enter image description here In the next image below: The car corners on the side of a large rotating turntable. The wheels now leave travel paths behind that are not circular. No longer is there a turn center, yet the wheels still leave behind contact travel paths that are circular just as it would when the turntable is still.

This is evidence that a car is not in a state of rotation around a turn center when cornering. The reason a car travels a circular path when the front wheels are turned is found in a simple Cornering Mechanism that is common to most self-guided vehicles. enter image description here

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3 Answers 3

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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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EDIT: Decided to summarise the lengthy argument below, with a much shorter mathematical argument here. The turning radius of the front wheel ($R_F$) at any instant is given by $$R_F = \frac{L}{\sin \theta}$$ where the steering angle $\theta$ is the angle of the front wheel relative to the rear wheel and L is is distance between the front wheel axle and the rear axle. If the handle bar is held in a fixed position, $\theta$ remains constant and $R_F$ at the next instant remains the same. If the steering angle is continually increased, then the radius of the turn reduces until at the extreme, the front wheel is at 90 degrees relative to the rear wheel and the turning radius becomes $$R_F = \frac{L}{\sin \theta} = \frac{L}{\sin (\pi/2)} = L$$
and the turning centre is now under the rear wheel. The turning centre is never under the front wheel.

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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)

It does not change the way it changes direction.

At any given instant, the front wheel always follows a curve with radius R relative to its instantaneous turning point or centre, where R is the distance from that turning centre.

While the handle bar is turning the wheel, the location of the instantaneous turning centre is continuously changing and while the handlebar is held at a constant non zero angle, the location of the turning centre is constant.

The wheels always follows a path that is tangential to the radii drawn to where their common turning centre is at that instant.

The instantaneous location of the turning centre is always the intersection of the perpendiculars to the tangents of the wheels at their contact point, as depicted in figures 1A to 1D of the original post.

What if, for the red path segments, the wheel was rolling freely - not attached to the bicycle - and someone was pushing the wheel axle to make it turn and roll along that red path? Would it then be changing the direction it rolls as a result of the rotation around a vertical axis that goes through its contact patch? I would say yes. – Matt Zusy

You raise an interesting point. I guess there is a case for the argument that a wheel that remains upright would turn around a point directly under the wheel if the wheel is being forced to turn by external forces and if the wheel has no forward motion. However, consider the case where the wheel has forward motion and you manually force the wheel to follow a circular path by continually turning the direction the wheel the heading in. That circular path would have a centre and that would be the turning centre of the wheel and it would not be directly under the wheel. There are two separate issues at play here. On is the change in angle of the front wheel relative to the rear wheel as the handlebar is turned and this change in orientation of the front wheel does indeed occur around a vertical axis passing through the contact point, but this is a separate issue to the path followed by the wheel as it moves forward where is has a turning centre that is not under the contact patch.

For a single independent leaning wheel with no external forces, the turning centre is determined by the intersection of the axis of rotation of the wheel and the surface the wheel is rolling on. The more the wheel is leaned over, the smaller its turning radius gets. This can be observed by rolling a tilted coin on a flat surface. It follows a nearly circular path but as it slows down its tilts more and the radius decreases so that its path spirals inwards. In the diagram below, the blue circle shows the path the back wheel follows, if it is not connected to the front wheel and the green circle shows the path the front wheel follows, if it is not connected to the back wheel. They each have independent turning centres (B and F).

enter image description here

When the wheels are connected by a frame, they are forced to share a common turning centre (O) and its location is determined as described earlier or by simply extending the lines passing through the contact points and the individual turning points to find their common turning point at the intersection of these lines. Now both wheels follow concentric circular paths as shown by the black paths in the diagram.

And when you’re not turning it, how is it turning? – Matt Zusy

If the front wheel is at a fixed angle relative to the rear wheel, then when the bike is pushed forward, the path of least resistance is the one in which the front wheel is rolling around its axle. If you push the bike in a straight line while the front wheel is turned, there is much greater friction between the tyre and the road, than when you let the front wheel follow a path that allows it rotate around its axle.

for instance, what if the (wheel was) rolling freely .. and gravity was removed. We would think the new wheel velocity direction is parallel to the wheel plane as the wheel continues to rotate around the axle. Now quickly turn gravity back on and the wheel starts turning again.. - Matt Zusy

While gravity is present there is a torque due to the mass of the wheel and there is a gyroscopic reaction torque which causes the wheel to turn and change direction. When the gravity is removed there is is no initiating torque to cause a gyroscopic reaction. This can easily be seen by suspending a rotating bicycle wheel by a string attached to one side of the axle only. The wheel precesses around the vertical axis. When another string is attached to the opposite side of the axle the wheel stops turning around the vertical axis, because there is no initiating torque. Your thought experiment neatly demonstrates that the turning of a front wheel when leaned over in a gravity field, is due to gyroscopic forces.

A vehicle wheel changes direction by forces at its contact patch only.. – Matt Zusy

We appear to talking past each other. I have mainly been talking about geometry of the path followed by a turning wheel and where the instantaneous centre of that circular path is located. You are talking about the location of where the forces act. These are two different things. In a gravitational field, the forces acting on a free leaning wheel are gyroscopic torque and forces acting at the contact patch. The geometric path followed by the wheel has a radius and centre far from the contact patch.

If you have the skill to ride a bike with holding the handlebars you will notice that when you lean to the right, the front wheel will automatically turn to the right.

Consider when you ride a bike straight and lean it. The wheels do not turn nor try to turn.. – Matt Zusy

Hold a bike at an angle to the vertical with the wheels parallel to each other and push it forward with no rider. I guarantee the front wheel will turn in the direction of the lean and the bike will follow a circular path. (I used to do this out of curiosity as a kid.) In order for a bike to travel straight while leaned over, a counter weight (e.g. the rider leaning in the opposite direction to the lean of the bike) is needed so that the COM of the combined mass is above the line joining the contact patches. Now the torque due to gravity acting around the horizontal axis is removed and now there is no gyroscopic reaction force, so the bike continues in a straight line while leaned over. (Guess who used to do this as a kid :-)

An additional subtle feature of a turning bike is that the contact patch of the front wheel moves from being directly under the axle when the bike is upright to a point significantly further forward as the bike is leaned over.

This phenomena is described in detail in my answer to "How does tilting a bike make it turn sharper?"

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  • $\begingroup$ Thanks @KDP! What if, for the red path segments, the wheel was rolling freely - not attached to the bicycle - and someone was pushing the wheel axle to make it turn and roll along that red path? Would it then be changing the direction it rolls as a result of the rotation around a vertical axis that goes through its contact patch? I would say yes. Then I ask, why does it change the way it rolls on the blue paths when it is attached to the bicycle? $\endgroup$
    – Matt Zusy
    Apr 2 at 23:59
  • $\begingroup$ @MattZusy I have extended my answer to cover the interesting point you raise. $\endgroup$
    – KDP
    Apr 4 at 4:59
  • $\begingroup$ Thanks again @KDP! Well presented. I just see things a little differently by considering other relevant phenomena. For instance, what if the blue or green wheel were rolling freely -not attached to anything- and gravity was removed. We would think the new wheel velocity direction is parallel to the wheel plane as the wheel continues to rotate around the axle. Now quickly turn gravity back on and the wheel starts turning again. This link explains why How bicycles… $\endgroup$
    – Matt Zusy
    Apr 4 at 12:09
  • $\begingroup$ Remember the surprise when you first learned that to turn on your bike, you steer in the opposite direction of the turn? Some things are out of our awareness until we look at it in a different way. A vehicle wheel changes direction by forces at its contact patch only. How the force direction changes is explained here. Cornering Mechanism $\endgroup$
    – Matt Zusy
    Apr 4 at 12:13
  • $\begingroup$ Further to my blue and green wheels comment: What you describe is a rolling cone with the cone points at points B and F. I don't think that is the case. Consider when you ride a bike straight and lean it. The wheels do not turn nor try to turn. I think a rolling falling wheel changes direction because its velocity direction is at an angle to the wheel plane. The “bicycle” link explains the rest. $\endgroup$
    – Matt Zusy
    Apr 4 at 13:17

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