01-12-2005, 07:53 PM
|
#213 (permalink)
|
| Guest | Re: Rim brake heat and spoke tension
In article <87is645v6u.fsf_-_@k-online.com>,
Joe Riel <joer@k-online.com> wrote:
> 1/Feff = 1/Fs + (n/2/pi)/Fr
This is, I would say, the most important single equation of your
analysis, and represents the meeting of the spokes' spring constant with
the equivalent constant presented by the rim - everything else is just
baby steps of definition. But, you skipped over a very interesting
number - this constant for the rim - which should give some idea of the
relative magnitude of what happens in the rim vs what goes on with the
spokes.
I'll use your cross sectional areas (2mm^2 for the spokes,
80mm^2 for the rim) but for my own sanity, I'll keep forces in N and
use metric values for material constants.
Fspoke = 2mm^2 * 200GPa = 400kN
Frim(eff) = 80mm^2 * 70GPa * 2*pi/36 ~ 977kN
This suprised me somewhat: for a uniform radial load across the
entire rim, the rim's share of that load is more than twice that of the
spokes! This is quite different from the intuition Jobst gives us of the
spokes in the LAZ taking all the load with the rim serving as a
pretensioning element for the spokes.
I still need to think a bit about what's going on, but I don't
see this as necessarily contradicting anything - the 'spokes do all the
work' intuition is for when the load is on a couple of spokes. Increased
rim compression is a secondary effect, but accumulates as you add
loads around the rest of the circumference.
It'd be interesting to take another look at Jobst's FEM results,
but my copy of the book is three time zones away.
Another interesting excercise would be to look at the relative
effective yield strengths are for each. ISTR the tensile yield strength
of spoke steel to be around 1GPa, while 7178-T6's compressive yield
stength is around 530MPa according to:
http://asm.matweb.com/search/Specifi...ssnum=MA7178T6
Yspoke = 2mm^2 * 1GPa = 2kN
Yrim = 80mm^2 * 530MPa * 2*pi/36 ~ 7.4kN
This would suggest that in tensioning a wheel, the spokes would
yield way before the rim does! Something doesn't seem right to me,
but, I have no idea how representative the yield strength I dug up
would be for a typical rim, nor do I have a good feel for how good the
80mm^2 estimate of the sectional area is (though it doesn't strike me
as being too far off).
Hmm... for comparison, a 6061-O alloy at:
http://asm.matweb.com/search/Specifi...assnum=MA6061O
lists a yield strength (though tensile) of 55.2MPa, an order of
magnitude less than the 7178. This would have the rim yielding ~
..77kN, well before the spokes. This seems more reasonable.
>So a 100degC rise in the rim temperature increases the spoke
>tension by 75kgf (165lbf). This is not a trival amount, it
>represents about 25% of the ultimate strength of a spoke.
I get to pretty much the same tension too (though for the
yield strength I worked out, it's an even scarier 38%, so if there's an
error in your reasoning, I've made the same folly.) I should point out
though, this increase in tension doesn't happen alone - when you have
an inflated tire on the rim, the tire pressure on the rim bed can be
pretty significant:
622mm*pi/36 * 20mm ~ 1.7 in^2 for a 20mm wide rim
With a tire inflated to 100 psi, this would be 170lbs of
pressure per spoke. But, while this number is remarkably close to the
tension increase we got earlier, this force is supported by both the
rim and spoke, of which the spoke sees:
170 * 40/(40+97.7) ~ 50lbs
with the other 120 being supported by the hoop strength of the rim.
Offhand, I'm not sure what the forces that a tire casing applies
to a clincher rim would look like and what it contributes to this
picture.
Other numbers aside, the 2mm^2 spoke area would be for a 1.6mm
diameter spoke. 15ga, which I'm more comfortable with would give a spoke
section of 2.5mm^2, and 14ga gives us 3.14mm^2. Going to these higher
gauges would change how the numbers balance.
-Luns
|
|
|  |