Undersampling?
Probably not.
Takeaway: don't reflexively scream "undersampling!"
If you are recording the time-resolved force for falls on ropes, how fast do you need to sample to get the peak value, within a reasonable uncertainty? Some climbing aficionados will claim you need to sample each millisecond (i.e., 1000 Hz).
Most high-end load cells have an inherent responsiveness that is less than a millisecond. They sample at less than 1000 Hz simply by reading force at selected times; a sampling rate of 100Hz just samples every 10 milliseconds. Tests that record the forces during falls (e.g., Fig 9a in Pavier (1998) Sports Engineering I, pp79-91) give peak widths (FWHM=full width half maximum) of more than 200 ms; 40 Hz would be enough to define the top of the peak with 8 points. However, those tests involved additional sources of damping (e.g. travel past “carabiners” and truly dynamic rope), so it is not clear whether those tests are exactly relevant to my situation.
I tend to get broader peaks with
falls on very static cords; those also hurt much more, even though the
peak forces are about the same. With very static ropes, less of the
fall force is absorbed in the rope, and more is immediately absorbed by
the deformation of my body.
My
situation!
I
do drop
tests of 24” to 32” on 10’ of double-strand cord and rope; the dropping
object
is me in my harness. Mainly I intend to check if a short fall can create shock loads that will snap the anchor sling.
A linescale
3 is at the upper end of 10’ rope/cord; the rope is simply passed through the
lower load eye of the linescale. I stand on the “base” step right below the
linescale, and adjust the harness knot till when I am hanging in the harness, I
can barely touch my feet flat on the step. Then I climb 3 or 4 steps up, and
jump up and out so I will fall more or less right under the load cell. Of
course I pull my legs up so I won’t smash into the ground. (The ceiling of my stairwell is massively reeforced.)
I jumped 2 times at 24" (3 steps) and once at 32" (4 steps) for the 8mm nylon. Here is the force vs. time plot; see below for a possible explanation of why last 24" jump came out close to the 32" jump:
Here’s a
close-up of the peak for the last jump:
And as you
can see, the peak is defined by just 6 points above FWHM at 40 Hz. Surely
that is undersampled?
Well
probably not. First consider what peaks should look like in this type
of test:
damped harmonic oscillators with a sine wave modulated by a
pre-function, which
is a decaying exponential in the simplest case. This general shape is consistent with Pavier (1998). The central part of the
peak will look
something like the center of the sine function at left below; the real
measurement will have tails in part because my setup doesn't allow or
measure negative forces. The worst case scenario (i.e. least likely to capture the central maximum) is when the six sample
points are evenly distributed between the ends of the FWHM, as in the plots below. At right below is a guassian, constructed to have the same FWHM as the sine curve.
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The true maximum of the sine function (left) is at 1.0 on the plot; the highest samples are at 0.978. For the gaussian, the highest sampled point is 0.973 (there is no reason to expect the peak would be gaussian). Given that this is the worst case, is ~2-3% sampling error significant?
I use a
somewhat abused linescale 3 to measure force in these experiments; the double
strand rope/cord simply loops over the lower load eye of the instrument (i.e
the rope isn’t tied to the linescale). We don’t measure negative forces,
because you can’t push hard on a rope, and if you did, it would simply slip
past the load eye.
What is a reasonable uncertainty estimate?
My linescale
is not great at measuring low-end forces. When I pick up a 30 lb weight, the
reading drifts by 6 lbs over a 20 second interval, and the drift in the
absolute zero is about 10 lbs when the test involves several force excursions.
So when the peak measurement may be low by 12 to 18 lbs = 2% to 3% of 600 lbs (in the worst case),
and the
inherent drift of the baseline is 10 lbs, I care not to gild the lily.
An even greater source of uncertainty arises from the difficulty of reproducing
any experiment with so many
parameters to control. Those parameters include the
angle and height of each jump, the configuration of my body, and
especially, tightening of the knot. Consider that the first two jumps
were supposed to be replicates,
but differed by 120 lbs. In review of the video, I can see that I
jumped higher for the second test, so it was probably more than 24"; by
I had no awareness of this at the time. In addition, the knot was just
tightened under body weight before the first jump, and the knot offered some
energy absorption as it tightened under the first fall. Simple knots like the figure 8 (used in this test) can absorb > 20% of the energy of a fall (Martin et al. (2015) J. dynamic behavior mater. 1, pp214-224.)
Why don’t
I use a higher sample rate? The only options of the linescale are 10, 40, 640,
and 1280 Hz. I started out using 640 Hz. If you have ever tried to save logs
from the linescale, on a test that involves multiple force excursions, you will
understand why I opt for 40 Hz. Even at 40 Hz the csv files can be long and are
saved as buffered, overlapping pieces. The test above involved 3 overlapping files, and
one must search through the csv files to find the overlaps so the data can be
pasted together into one spreadsheet. On my instrument, the data-out usb port has
become deformed and must be held in place forcibly while I’m downloading the
data, so I have to make sure the files are reasonably short. (Note the smart
phone app is limited by usb connection speed, and is capped at 40 Hz.)