In theory, we can measure spoke tension if we know how long the spoke
is, hang a known weight from its midspan, and measure the spoke's
downward deflection:
tension = (length x force) / (4 x deflection)
But this measurement can be tricky to perform on a real wheel.
So I put a spoke in a vise-rig and hung two weights from its midspan
by thin wires. I also laid a thin ruler flat across the vise jaws to
serve as a reference plane when I measured the small deflection.
I used Post Office electronic weight scales and dial calipers.
Here's a view of the setup:
http://i17.tinypic.com/4bf8whi.jpg
Click on the lower right in Explorer for the full-size image.
The camera angle isn't dead level with the thin metal ruler laid a
across the jaws, but it's good enough to show the 19 lbs of weight
pulling the spoke down.
The length of the span between the vise jaws was 4.545", the two
weights together weighed 19 lbs 2.5 ounces (19.15625 lbs), and the
downward deflection was a suspiciously round 0.120", so I've
calculated the tension for 0.110, 0.120, and 0.130 inches.
(4.545 x 19.15625) / (4 x 0.110) = 212 lbs tension
(4.545 x 19.15625) / (4 x 0.120) = 195 lbs tension
(4.545 x 19.15625) / (4 x 0.130) = 180 lbs tension
Whatever the spoke's original tension was, it certainly looks as if it
must have risen to around 195 lbs.
But despite my faultless theory and careful measurements, a small
correction seems to be needed to get the correct spoke tension:
http://i18.tinypic.com/30c6a3c.jpg
🙂
A real wheel may have a few smaller confounding factors hidden in it
that aren't as easily revealed.
***
So much for deceiving trusting readers as an object lesson. There are
no hidden flaws in what follows--at least no _deliberately_ hidden
flaws. Like anyone else, I may be blind to what's wrong with my
explanations.
Let's consider the relative accuracy of measuring deflections in real
wheels versus a Park gauge.
We measure a spoke in a real wheel as 280 mm from hub to rim.
(Let's use a radial wheel, so that we don't have to wonder about that
pesky bend at the spoke crossing.)
We hang a 23-pound weight from our 280 mm spoke's midspan and measure
the downward deflection as 7 mm.
Then we add a 26-pound weight and measure the deflection again, this
time as 12 mm.
We apply our spoke tension equation again:
tension = (force x length) / (4 x deflection)
(23 x 280) / (4 x 7) = 230 pounds of tension
(49 x 280) / (4 x 12) = 286 pounds of tension
There's no deflection with 0 pounds of weights, we can't tell what the
original tension was.
But it looks like a 56-lb tension increase when we add 26 lbs of
weight, so we seem to be getting two pounds of tension increase for
each pound of side force.
But this is considerably more than anyone has yet reported measuring
with a tension gauge on a real wheel.
So we simply dismiss the Park gauge as inaccurate.
But what's the margin of error in our measurements versus the Park
tool?
Again, let's ignore the length. It's 280 mm. And let's assume that our
23 and 26 pound weights total exactly 49 pounds.
But it can be tricky to measure the deflection of a spoke with a rope
or wire wrapped around its midspan in a real wheel.
Maybe the deflections really are 7.0 mm and 12.0 mm on the nose, but
what if those were a little off, say plus or minus half a millimeter?
280 mm, 23 lbs
deflection tension change
6.5 mm 248 lbs +18 (280 x 23 ) / (4 x 6.5)
7.0 mm 230 lbs 0 (280 x 23 ) / (4 x 7.0)
7.5 mm 215 lbs -15 (280 x 23 ) / (4 x 7.5)
280 mm, 49 lbs
deflection tension change
11.5 mm 298 lbs +12 (280 x 49) / (4 x 11.5)
12.0 mm 286 lbs 0 (280 x 49) / (4 x 12.0)
12.5 mm 274 lbs -12 (280 x 49) / (4 x 12.5)
Hmmm . . .
If the margin of error is +/- 0.5 mm, then our first tension is really
215~248 pounds, while our second tension is 274~298 pounds.
That would mean that the tension rise for adding the second 26-pound
weight is between 26 pounds (274-248) and 83 pounds (298-215), or
anywhere from 1-to-1 to 1-to-3.
After measuring our single spoke, we declare that the test gives us
what we wanted, that it proves that literally hundreds of spoke tests
on other wheels with a Park gauge that showed nothing better than
1-to-1 ratios must be wrong, and that our wheel must be much stiffer
than all those other wheels.
Stiffer, in fact, than a massively braced pipe-clamp rig with no spoke
crossing that achieved only a 1.5-to-1 ratio when measured with a Park
tension gauge.
Hmmm . . . let's take a look at that inaccurate Park tension gauge.
The marks on a Park tension gauge are spaced at 0.075 inches, or 1.905
mm. When applied, the Park gauge hangs as motionless as the spoke and
is fairly easy to read at a glance.
So a quarter mark on the Park tool is about 0.5 mm.
Repeated applications of the Park gauge will produce readings
consistent to within a quarter-mark or better, so a margin of error of
a quarter-mark either way seems reasonable. (More expensive tension
gauges may do even better.)
For a 2 mm round steel spoke in the 200-300 pound range (91 to 136
kgf), the Park gauge table reads like this:
Park Park extrapolated
mark kgf lbs
22.00 85 187
22.25 193
22.50 198
22.75 203
23.00 95 209
23.25 215
23.50 222 [censored]
23.75 228 +/- ~7 lbs @ ~230 lbs
24.00 107 235 [censored]
24.25 243
24.50 251
24.75 259
25.00 121 266
25.25 275 [censored]
25.50 284 +/- ~9 lbs @ ~280 lbs
25.75 293 [censored]
26.00 137 301
So the Park gauge looks accurate to +/- 7~9 lbs at the expected
tensions if read to within a quarter mark, about 0.5 mm on the scale.
But an error of the same +/- 0.5 mm on our single deflection test,
which was much harder to measure, meant an accuracy of +/- 12~18
pounds--about twice as large a margin of error.
Yet the argument is that the Park tool must be less accurate . . .
Even though it accurately measures a 190-lb weight hung from a spoke
and comes with a factory calibration stamp, with a handwritten date
and the inspector's initials:
http://i10.tinypic.com/47i07xv.jpg
Cheers,
Carl Fogel