Car tire tracks while turning in a circle

[Charlie Roberts]

Does anyone have pointers to any *good* documentation
(in order of preference, papers, presentations and videos)
of the tracks (or, path) of a car's tires as it turns in a
circle (the steering wheel kept at a constant angle).


Spending about two hours on Google, yielded a couple
of hits, but the material was not very well done. I did
pick up a number of other things (like the Ackermann
steering mechanism) that I did not know about, but I
could not find a simple analysis. What I would like to
see is

1. A simple case of a four wheeled vehicle going in
a circle without the complications of wheels slipping,
axles under strain, etc. For a specifc case, let the
drive wheels be the rear wheels and the front
wheels make the *same angle* with respect to
the perpendicular to the vehicle's instantaneous
direction of motion.

2. The case when the rear wheels can also be
turned, but not necessarily at the same angle as
the front wheels.

3. Any other complication, like wheel slippage, is
okay, but not essential.

What I am really after is a for a way of calculating
the radius of the circle give the angles of the
wheels (all four) with respect to the longitudinal
axis of the car (i.e. the instantaneous direction
of motion) given the tread (distance between
the two rear or front wheels, assumed to be
the same) and the wheelbase (distance between
the front and rear wheels, assumed to be the
same).

The closest that I got to what I was looking for
is this:

If there is a better source, please let me know.

tia

[John Francis]

Post by Charlie Roberts
Does anyone have pointers to any *good* documentation
(in order of preference, papers, presentations and videos)
of the tracks (or, path) of a car's tires as it turns in a
circle (the steering wheel kept at a constant angle).

That's fairly simple.

All the wheels move in circular tracks around a common point.
This point is somewhere on a continuation of the rear axle.
The steering linkage of the car is designed such that the
directions in which the front wheels point changes by slightly
different amounts when the steering wheel is turned; the
intersection of (a continuation of) the two axes of rotation
of the front wheels falls on a continuation of the rear axle.
[This isn't 100% true; there's no way that a simple linkage
such as that found in car steering systems can do that exactly,
but it is close enough; sideways slipping of the two front
wheels can take care of the small errors the linkage causes]


To look at this another way:

Choose a center of rotation somewhere along (the continuation of)
the rear axle of the car.

Draw lines from this point to the centers of the two front wheels.
Each of the front wheels should be pointing in a direction that
is perpendicular to this line (which is what the steering linkage
is designed to do).

[Charlie Roberts]

Post by John Francis

Post by Charlie Roberts
Does anyone have pointers to any *good* documentation
(in order of preference, papers, presentations and videos)
of the tracks (or, path) of a car's tires as it turns in a
circle (the steering wheel kept at a constant angle).

That's fairly simple.
All the wheels move in circular tracks around a common point.
This point is somewhere on a continuation of the rear axle.
The steering linkage of the car is designed such that the
directions in which the front wheels point changes by slightly
different amounts when the steering wheel is turned; the
intersection of (a continuation of) the two axes of rotation
of the front wheels falls on a continuation of the rear axle.
[This isn't 100% true; there's no way that a simple linkage
such as that found in car steering systems can do that exactly,
but it is close enough; sideways slipping of the two front
wheels can take care of the small errors the linkage causes]
Choose a center of rotation somewhere along (the continuation of)
the rear axle of the car.
Draw lines from this point to the centers of the two front wheels.
Each of the front wheels should be pointing in a direction that
is perpendicular to this line (which is what the steering linkage
is designed to do).

I get that ... there is all that stuff about the Ackermann steering
mechanism and all that. My question in more fundamental. How
do we get the path of the car given that the rear wheels propel
the car along its length. I suppose it is the same with a boat,
with the rudder kept at a constant angle.

The next step is what is the path/tracks if the steering angle
is a function of time.

I do not see any equations that derive the path.

There is also an interesting YouTube video on bicycle tire
tracks. Again, no derivations.