From: zowie@daedalus.stanford.edu (Craig "Powderkeg" DeForest)
Newsgroups: rec.motorcycles,sci.physics
Subject: PHYSICS OF STEERING (long) (was Re: What would you ride on a long...)
Date: 19 Nov 92 02:07:25 GMT
	<1992Nov18.020307.19538@tcsi.com> <BxxoMr.F6@news.iastate.edu>
Organization: Stanford Center for Space Science and Astrophysics


It's about time a trained physicist posted something about this...
Here's rec.moto Steering Argument #597.

In article <baz> tomes@iastate.edu () writes:
   In article <bar> markk@tcs.com (Mark Kromer)writes:
   >At turn in, the motorcycle rotates about the cg not the contact
   >patches.  Re: countersteering.  Low polar moment in the transverse
   >plane (and in the longitudinal plane but for different reasons) is the
   >key, not low cg.  Keep the mass concentrated as close to the cg as
   >possible.

   Ok, I used the wrong term (polar moment).  But show me how an object
   can pivot about a point above the ground and still remain in contact with
   the ground (assume point touching the ground is the point farthest away
   from the CG in that immediate part of the object eg contact patch).
   The instantaneous roll axis for a singletrack vehicle MUST be the
   line between the two contact patches; otherwise the bike is off the ground
   by definition.

Okay.  Basically, by moving the center of rotation downward.

The point (pun intended) here is that the cg moved (by gravity) downward by
gravity, to maintain the tire's contact with the ground, during the roll
manoeuver.  There's no contradiction here -- for example, (see a different
thread) when you're driving in a straight line, your front wheel is rotating
about a single axis, the center of the front axle.  But that axis (here's the
whole point of owning the machine!) is _moving_ at the time.  Linear motion
and rotation are related but separate.

Here's a cheesey ASCII diagram of a mcy doing a *stationary* right roll, as
seen from a guy equipped with an ASCII camera, immediately behind it.  The
mcy was just sitting there upright, and someone breathed on it and it started
falling to the right.
                     	         Here, I've marked the cg with an asterisk
           |                     (Little johnny started his bike/upon the
      	   |   	   /   	         road to frisk./What a stupid guy he was,/
      	   |      /              his little *)  The thing to notice here is
       	   *     /               that the c.g. went *down* as well as moving
     	   |    *   <--- c.g.    a bit (I've drawn a dotted line to the
     	   |   /:                leaned position, caught in mid-fall).
     	   |  / :
     	   | /  :
     	   |/  	:
 ======================ground

When a stationary motorcycle falls, the contact patch is, by definition,
stationary -- it doesn't move.  Gravity pulls straight down on the c.g.
of the bike, and the ground pushes straight up on the contact patch.
The bike experiences a roll torque proportional to the horizontal distance
between these two points -- so, if you could only balance the bike _perfectly_,
it'd never fall.  But the mere flap of a butterfly's wing would blow it over.

When you ride, you (and, in anything that's not a total cafe racer, the
designers of your slanted front forks) are steering the front wheel so as
to keep the contact patches directly under your c.g., so you don't fall over.
If you don't do this, you will fall over.  Try this on a bicycle (*NOT* a
motorcycle): tie the handlebars in a straight-ahead configuration, with some
twine.  Try to ride and balance the bike.  You can't -- the dynamics of the
moving case are the same as the stationary case, if you don't turn the
handlebars.

When you steer, you apply a horizontal force to the front contact patch, by
steering the front wheel. This exerts a roll torque on the bike, since the
bike's inertia opposes the applied force (action, reaction, y'know) from the
c.g.  The bike feels a torque proportional to the force you exert by steering,
times the vertical distance between the contact patch and the c.g.
It rotates, not about the c.g. *or* the contact patch, but about the point
half-way in between them.  Here's a diagram:

           |
           |                             /
           |	    (contact    	/
           |        patch gets         /
           *	    steered to        /
           |	    the left)        *
           |		            /:
           X		           x :
           |                      /: :
           |		         / : :
  =========+==========    =========+=========

     BEFORE STEERING	    AFTER STEERING

Note that the center of rotation (c.r.), here marked with an X, has
moved _down_ a bit, as has the c.g.  (I tried to make it look lower by
making it lower-case, but it really should be 1/2 line lower...)

You had to steer the front wheel *left*, to make the bike lean *right*.
Then, to keep from falling over, your trained reflexes, helped a lot by
gyroscopic precession (see the next chapter :-) and the shape of your front
fork, steer the front wheel to the *right*.

When you steer the front wheel to the right, you apply a rightward force to
the contact patch, and the bike responds with a reaction force to the left,
from the c.g.  The bike feels a roll torque to the left.  When you're holding
a smooth right turn, the roll torque to the left from the tires exactly
balances the roll torque to the right from gravity.  Here's yet another
diagram:


                       /
                      /
       	       	     /
       `centrifugal /
        force' <-- *
                  /
                 / | Gravity
              ^ /  V
       Support|/
       ========= --> steering force ======

The idea here is that the centrifugal force and gravity acting on you and your
bike, add up to a force along the line from the c.g. to the contact patch,
otherwise the bike would fall over.  The contact patch exerts (you hope!)
whatever force on the road is necessary, to keep from sliding on the road.
The upward part of the force you get for free; the sideways steering force is
why you buy expensive tires with sticky, quick-wearing rubber.

This is why it never feels like you're turning, on a bike -- when you're in a
cage, mountain roads throw you around; yawing the vehicle makes `local down'
change towards the side of the vehicle.  When you're on a bike, you *have* to
roll the vehicle so that local down points between the c.g. and the contact
patch; otherwise, you'd fall over, like a bike with the handlebars tied
straight.  You feel a force straight down the direction of your spine into the
seat of your pants, rather than a sideways force that knocks you over.

Incidentally, you can ride a bicycle no-hands because you constitute most of
the mass of the rider-vehicle combination.  Moving yourself around, causes the
c.g. to move significantly, and the whole rolling process happens (with the
gyroscopic force on the wheel, and the design of the forks, conspiring to keep
you near equilibrium).  On a motorcycle, the vehicle outweighs you, so the
effect of leaning sideways is minimal -- the c.g. only moves a little bit, and
the bike doesn't turn (yaw) very much at all.  Those of you who are
{adventurous | stupid} have already bought throttle clips and ridden your
motorcycle no-hands on a straight, level road at some point.

More quotation:
   (This is bad:  A student pushed much too hard when
   countersteering in a class and threw the bike out from under
   himself.  I saw the front wheel airborne for a fraction of a second while
   the bike was busy pivoting about the CG, like you suggested.  I believe this
   means that there is an upper limit on lean-in rate that is determined by
   the acceleration due to gravity and the CG height above ground.  If one
   increases lean angle too quickly, the bike rolls about the cg and not about
   the contact patches.)

Good call -- what went on here is exactly that: the force of gravity couldn't
accelerate the c.g. *down* fast enough to match the roll of the bike.  In a
weightless environment, the c.r. (1/2 way between the contact patch and the
c.g.) would be stationary during the roll; gravity (almost) always pulls the
c.g. down fast enough to maintain the contact with the ground; when it
doesn't, you can't very well roll out of the turn, can you?

   BTW, the same argument I used to defend [Honda Gold]Wings applies to polar 
   moment also: since most of the engine/trans lies within a foot of the axis 
   passing through the cg and parallel to the line between the contact 
   patches, the bike handles better than you would think if all you 
   considered was mass.

Yep -- bikes, and engines (most of the mass of the bike :-) that cluster
around the c.g., make it easier to steer, than engines that spread out far
above/below it.  Though I believe that, in most cases, steering geometry
(wheel size, fork angle, offset between forks and axle, length of wheelbase)
affects handling *much more* than does mass distribution in the bike.  The
reason I think so is twofold: (a) unless you're screaming along around the
twisties, you're not rolling the bike very much compared to the acceleration
due to gravity -- you just don't lean very fast; and (b) *you* don't have
to apply the force to turn the bike -- the tires do that for you.  Almost 
any steering force you have to apply, was designed in by the engineers who
drew the plans for your bike.  If they wanted effortless turns, they'd 
place the contact patch closer to the (fork-line, ground) intersection;
if they wanted a bike that liked to cruise in a straight line, they'd place
it far behind that intersection.

BTW, in a very tight turn the top half of you (the rider) is significantly
closer to the center of rotation of the yaw, than is the bottom of you.
This means that `down' is tilted more, relative to the horizon, for your bike
and your feet, than it is for your head, so you might notice your back swaying
a little while your balance tells you your straight!  [Don't tell novice
riders/passengers this though, or they'll use it to justify keeping their
bodies oriented to the horizon :-(]

--
Craig DeForest -- astrophysicist for hire.  

Beat Cal - Beat Cal - Beat Cal - Beat Cal - Beat Cal - Beat Cal - Beat Cal 
               [sort-a mindless, eh? That's pac-10 for you.]
Newsgroups: rec.motorcycles