1. Forum
  2. Usenet
  3. REC.PUZZLES

  • Car tire tracks while turning in a circle

    From Charlie Roberts@21:1/5 to All on Sun Jan 22 15:28:46 2023
    XPost: alt.math.recreational

    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:

    https://www.youtube.com/watch?v=i6uBwudwA5o

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

    tia


    --
    This email has been checked for viruses by Avast antivirus software. www.avast.com

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From John Francis@21:1/5 to croberts@gmail.com on Mon Jan 23 01:19:44 2023
    XPost: alt.math.recreational

    In article <pg5rsh9sofoijq800gtp24dro807dnkebi@4ax.com>,
    Charlie Roberts <croberts@gmail.com> wrote:

    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).

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Charlie Roberts@21:1/5 to Francis on Mon Jan 23 19:44:27 2023
    On Mon, 23 Jan 2023 01:19:44 -0000 (UTC), johnf@panix.com (John
    Francis) wrote:

    In article <pg5rsh9sofoijq800gtp24dro807dnkebi@4ax.com>,
    Charlie Roberts <croberts@gmail.com> wrote:

    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).

    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.

    --
    This email has been checked for viruses by Avast antivirus software. www.avast.com

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)