Re: shifting one's weight

[palmer@sfu.ca (Leigh Palmer)]

James McLean proposes a reasonable system related to the bicycle problem:

> As promised, here's a simple model for shifting one's weight:
>
>                __
>               |  |-----X
>               |  |
>         ______|__|___________
>        |_____________________|
>                  /\
>                 /  \
>                /    \
>
> That's a see-saw made of a board on a pivot.  Note that the pivot is
> constrained to stay in the center of the board.  The 'X' is a significant
> mass, and it is held on an arm which is capable of extending due to some
> internal energy source (that could be a compressed spring overdamped by a
> dashpot, for instance).  The mechanism is placed so that it's center of
> mass is above the pivot, so that the whole thing is in metastable
> equilibrium.

Good setup. This is what is called unstable equilibrium. The term
"metastable" should never have been invented. Any displacement from
equilibrium here, however small, results in an unstable system. It is
entirely analogous to the bicycle at rest with the headset clamped.

> Now extend the arm.  Internal forces cannot move the center of mass of a
> system.  Therefore, if the board were on a frictionless surface, the mass
> would move right and the board would move left.  But the board is
> constrained not to move: the pivot exerts a frictional force on the board
> to the right.  This external force pushes the center of mass of the (board
> + mechanism + mass) to the right, allowing gravity to then pull the board
> down on the right.  This is "shifting one's weight.

In the case of the bicycle one must neglect the mass of the bicycle
itself. Many riders (myself included) prefer a bicycle with the lowest
practical mass because a low mass bicycle feels "better" to us. My
ideal bicycle would be massless. In any event the mass of the bicycle
is less than that of the rider (less than 1/7 in my case, but I don't
have a particularly low mass bicycle for economic reasons). This is,
I suggest, the principal difference between the bicycle and motorcycle
cases.

Pursuant to making your example more nearly analogous to the bicycle
you should shorten the seesaw to a length equal to the pedal
separation on the bike, and locate the feet to either side of the
fulcrum. This will reduce the moment of inertia of the board itself.
The moment of inertia of the board is of considerable importance in
balancing a see-saw, as I'm sure you are well aware. That is why
daredevil wirewalkers carry poles as long and heavy as they can
manage. They don't seem to realize that their poles could be more
effective long, strong and light, with weighted ends. A suitably
loaded vaulting pole would be ideal.

> One feature of this model which connects to my experience, and therefore
> convinces me that it's basically correct, is what happens to the board.
> Note that, considering the (board + mechanism) alone, it experiences a
> torque out of the page, so that it initially tips down on the *left*.
> However, angular momentum of the (board + mechanism + mass) must remain
> zero, so that once the extension has occured everything will tip down on
> the right as described.

That's a motorcycle see-saw. with only two feet on it you can't exert
a torque on an inertialess see-saw.

> This initally came up in the context of what is going on in bicycle
> riding.  If you replace the pivot with the road, the board with a bike, and
> the (mechanism + mass) with a person, then the system above applies.  It
> remains to be seen whether the effect is large enough to actually platy a
> significant role.  Unfortunately, the initial torque on the board is
> important in determining how far the CoM moves.  On bicycle it is
> difficult to say just what the moving mass is, or with what lever arm it
> acts.  This makes any calculation speculative at best.

Replacing the pivot with the road is unrealistic for a bicycle. A
bicycle has an adjustable pivot. One can move the front of that pivot
from side to side by turning the handlebars. That is how one balances
a stationary bike. One does not "shift weight" from side to side of
the pivot; one shifts the pivot axis from side to side underneath the
rider's center of mass. Now that is for the ideal case, you must
remember. Even in the practical case, I don't think you will find
many riders who are capable of doing a track stand with a clamped
headset! It is possible to do so with a massive bicycle, but
impossible with an ideal bicycle, and practically impossible with a
good track bike.

> Instead, I suggest an experiment.  Ride a bicycle standing on the pedals
> (no seat contact) and only lightly touching the handle bars.  Now move all
> your weight onto one foot.  This morning I found that this had a very
> significant effect on the bicycle, leaning it over and tending to make it
> turn.

You will find that the results of this experiment depend sensitively on
the initial condition, the lean (or displacement of COM from line of
support) of the rider just before the weight is transferred. Try it
*both* with the bike in a very gentle turn to the left and to the right,
both times transferring weight to the left foot. I think you may be
surprised at the different results.

> PS.  While riding fast in a tight cicle (to check whether the front wheel
> is actually turned a bit), I also found out that it is possible to break a
> rear axle by skidding severely!  No damage to myself, happily.

On the flat?! You must be even heavier than I*. I used to break rear
spokes while laden with camping gear (always chainside, of course -
Murphy rides a bike), but I have never broken an axle.

In the final analysis it is the steering geometry that determines the
characteristic feel of what racers and avid tourists (I've been both)
consider to be an excellent bicycle. There are many misconceptions out
there, too. For example racers frequently consider that a bike with
good feel is a "stiff" bike, while nothing could be farther from the
truth. A perfectly laterally rigid bicycle would not feel right. What
they mean, I believe, is that the bike is elastic. When one deflects
the hanger to one side or the other, the energy is not dissipated, but
returns from storage. We physicists have a word for it, but the word
means something different (evoking images of rubber bands) in the
common parlance. Racers substitute "stiff". Once all these other
factors are made most nearly ideal (low mass, low wheel moments of
inertia, elastic response) the sole remaining factor is steering
geometry, and despite more than 100 years of attention to the problem
it is still the case that no one has succeeded in finding the magic
formulation that produces the ideal - but we know it when we feel it!

Who said there were no great unsolved problems left in physics?

Leigh

*or, perhaps, you have a hollow rear axle? Bad idea. Now you know
why. Carry a wrench instead, and install a solid axle.