Re: Hey, Jobst, on p39 of The Bicycle Wheel the graph appears to show the ?impossibility of...

On 2008-09-13, pm <zzyzx.xyzzy@xxxxxxxxx> wrote:
[...]
Stiffness is the ratio of force to displacement; on the graph, the
lateral stiffness of the wheel is given by the slope of the 'lateral
force' curve. I believe that it is evaluated for small displacements
around zero; not large(r) displacements that would cause spokes to
become slack.

Yes, I think you're right. It's just extrapolated as if the spokes
carried on working in compression.

I suppose the idea is that it's easier to see the gradients with the
full line there and so that it's steeper on the right, which is one of
the points being made. But it could be dashed on the left side past the
slack point.

Or perhaps show the real line with a discontinuity, as well as or
instead of the dashed line.

Now, about this curve. On closer inspection I do not see why the
lateral force line on the graph is drawn rectified; a leftward force
is opposite to a rightward force, yet the two forces are shown with
the same sign on the graph. If one were inclined to obstinate
shenanigans, one could try to trump this up into a "error" where the
graph shows the ?impossibility? of a rightward deflection to a
leftward force. Or one could read what is obvious from the salient
features of the graph. But I digress.

Yes, I thought that too at first.

But then I thought hang on, on this graph negative and positive force
are compression and tension, not leftwards and rightwards force. If they
were leftwards and rightwards force, then one or other of the T(L) and
T(R) curves would have to be drawn upside down.

Moving the rim left or right loads the whole structure with a shearing
kind of a force-- it compresses one side and tensions the other.

I think it's best to see the lateral force lines as two separate plots--
absolute leftwards force and absolute rightwards force.

But the caption does claim an interesting thing, that the wheel is
"twice as stiff for deflections to the right as to the left." It's not
immediately clear that this is true in the +- 4mm regime where equal
leftward and rightward deflections can be compared; one has to pull
out a magnifier and start counting pixels to see that the slope of the
lateral force curve approaching 0 from the left _might_ be less in
magnitude than the slope of the same curve approaching 0 from the
right.

I think that's why the left curve is extrapolated so far beyond 4mm-- so
that you can see the basic slope easier without getting out the
magnifier.
.