Member postings for S K

I don't think that's strictly true if the pendulum's shaft has a non-trivial mass compared to the bob, right? For example, if the pendulum was just a solid bar?

That was close to my situation, especially on the "light" side of the pendulum, in which the shaft and the small "bob" (the small pivot) were somewhat equal in mass. In any event, I wound up adding mass above the pivot when hung in the normal direction, which slows it down.

(But yeah, I spoke too loosely about it without explaining that the COO was actually above the bob, which is not typical for very heavy bobs.)

Edited By S K on 02/03/2023 19:37:45

Do you have any comments about the mechanical vs. your newer electromagnetic impulsing?

I developed a nagging suspicion that my last measurement of within 0.008% of the predicted g was a little too good, and too easily obtained to be that good.

After adding a base for the pendulum's support structure, I balanced the pendulum such that the normal and reverse periods were identical to better than a part in 1000. This reduces the contribution of the center of gravity measurement to a level making it nearly irrelevant. Note that when balanced in this way, the centers of oscillation of the pendulum are coincident with the pivots. Therefore, L in the ideal pendulum formula becomes the distance between pivots. And ... my new, more precise measurements did not improve the accuracy of g.

I then performed a basic error propagation analysis and concluded that my earlier finding was indeed more down to luck than actual precision. The problem is the measurement of L. For this, I used an engineer's scale (3 feet) with graduations of 0.01". How much precision can I get from this for measurements of about 2 feet?

For example, what value would you accord this example shot (just taken with my phone camera):

In this case, it seems clear: the anvil lies pretty much right on 20.83". Pixels are hard to see, but it's within one pixel of that, I believe. If it lay somewhere else, well, trust me when I say that interpolation is not easy, especially when trying to find the edge of a transparent surface. In this case, it's easier to see since it lies on a black line.

I also did tests that calibrated a much better camera and macro lens at a per-pixel level, to about 0.00045" per pixel, and I found that I could interpolate to about 0.002" with decent confidence (i.e. +/- 0.001" ). But this is almost certainly encroaching on the precision of the ruler itself (which I separately estimated to be within about 0.0013" over 35" ).

I did a basic error propagation analysis that showed that I'm almost completely limited by this measurement (the period T is measured to far, far greater precision). I've thus concluded that this project has measured g to all the precision that is currently possible for me to obtain:

g = 9.79 +/- 0.02 m/s^2.

This easily matches the estimate for g within those limits as well as g for the nearest known measured value.

I've thought about building a traveling microscope + glass scale that would likely do a little better. But it would be best if it's hung vertically with the pendulum rather than horizontally. Other than that or bringing the pendulum to NIST, that's all I can get.

(As an aside: With all due respect to Henry Kater, I have developed doubts about his claims, including that he measured L, also with just a ruler, albeit with a microscope rather than a macro setup, to a precision 100 times greater than the above.)

It was a fun project. 😀

Edited By S K on 02/03/2023 18:36:40

Oh, and does anyone have an answer to my pendulum bob orientation question?

I wonder what master-level clocks would have evolved to today if pendulums were still the most precise time-keeping technology?

I'd presume all the last developments prior to quartz oscillators (controlled vacuum, low COE materials, high-Q, small amplitudes, decoupled pendulums, etc.) but with electromagnetic drive and electronic measurement and compensation techniques much as folks here are playing with.

I'm still wondering how a synchronome-style mechanical gravity-impulsed drive compares to electronic drive. Certainly the best of the historical mechanical ones are still amongst the best of all. But I have to imagine that electronic drive can be made better due to the lack of friction if nothing else.

My next project will be a clock of sorts (probably still just a pendulum, with everything else electronic), but as I would not be aiming for ultimate accuracy, I've been toying with the idea of making a mechanical drive for it as a steam-punk flourish. Therefore, I've been wondering what a modern interpretation of that would look like. For example, electronic triggering is an obvious improvement over a mechanical counting wheel. And perhaps a servo might reset the drive weight rather than an electromagnet? Any other ideas?

That document is interesting.

After wading through (sometimes) politely-expressed but numerous long-standing grievances (what is it with old-time clock experts and their in-fighting? Or even modern ones - hehe.), we find that a pendulum in vacuum is some sort of over-balanced perpetual motion machine that can't be used to tell time! Air friction is a life-saver!

I guess geniuses are often "special."

Edited By S K on 01/03/2023 22:32:39

I have two simple questions about pendulums:

1) Given a typical disk-shaped bob, why is the vertical position (as typically seen in grandfather clocks) considered superior (e.g., "higher Q" ) to the same disk oriented horizontally (e.g., in my gravity pendulum)? One advantage of the horizontal position is that there's no possibility for a misalignment of the bob's angle of attack. It does push the mechanism out further, but what of Q?

2) To adjust a pendulum's period, a small weight is often added along its shaft. Sliding a weight (not adding a weight) lower on the shaft (closer to the bob) tends to slow it down because it moves the center of oscillation further down, while sliding it closer to the hinge tends to speed it up by moving the COO upwards. Leaving aside the mechanical issues, is there any reason why it's disadvantageous to increase the weight's effect further by moving it above the hinge (or below the bob)?

Thanks.

Edited By S K on 01/03/2023 17:13:21

I watched the whole video with interest (without the patience to work through his main paper). It's thought provoking, but some of it is contrary to hundreds of years of experience, and other bits (like invoking fractals) raised doubts (but I'm not enough of an expert to say exactly why).

I think I'd stick to high Q and minimal impulsing at BDC as the most likely path to success.

Edited By S K on 01/03/2023 17:15:09

You prompted some thinking: If I put it on a knife-edge, but had one end resting on a scale, and I measured the weight as I noted the position, when the weight just reached zero I'm in principle at the COG. If I then did the same in the other direction, I could plot the two lines and find an intercept. Interesting. A gram scale is probably not good enough, though.

Haha. I'm not in the U.K., but would you like my social security number and my mother's maiden name too? 😉

I'm working from an extremely fine-grained estimate. If I look at another house close to me, or change the elevation by the height of my workbench, the estimated number changes. So there would be an essentially infinite number of matching estimates, though with no matches at all in many areas.

I'd say 0.008% is pretty darned good, but I've got a few more digits of precision to go even to match Kater's claimed resolution.

As of now, I'm also 0.008% off the closest measured value that I know of, but that's about 60 miles away (I don't know the precise location). I expect I could find a spot where a high resolution measurement was made if I look a little harder.

I have a gram scale, but not sub-gram.

I redid the COG and distance measurements.

Finding and measuring the COG accurately is difficult. The usual technique is to balance it on a knife-edge. However, given the asymmetry in the pendulum and how long it is, there's a range of a few mm over which balance is arguably achieved, and there's some guess-work involved in settling on a final position. Even marking that position accurately is tricky (I was thinking I need some markup fluid and just scratch the shaft on the knife). If anyone has ideas on how to find the COG of a very awkward item with more confidence, please let me know.

Anyway, I redid all the measurements "blind" (without looking at my previous measurements). The anvil-to-anvil measurement turned out to be identical, but the COG position differed by 0.01".

I'm now within 0.008% of the predicted value of g at my location. 😀

Whoa! I wasn't expecting much, but for fun I made a quite rough measure of the COG and distances, and used my preliminary numbers from above (without equalizing the periods) and plugged it into the Repsold/Bessel formula.

I got g to within 0.026% of the predicted value. Not bad! 😀

Now to do a "proper" job of it!

Edit: The Repsold/Bessel innovation was to recognize that it's not strictly necessary to equalize the periods, as long as they are "close." This is because one term of their formula becomes quite small, leaving a term that only contains the two time periods, which I know extremely well. If, next, I can closely equalize them, then the problematic term (which requires measuring distances and the COG) becomes vanishingly small.

Edited By S K on 24/02/2023 19:52:29

Yes, I have a computed prediction for local gravity where I live, obtained from here:

https://geodesy.noaa.gov/cgi-bin/grav_pdx.prl

It does provide a (predicted?) error range, too. If I can match that within the error bars (I'm afraid that's unlikely), I'd be way more than satisfied! But one other possibility is to calibrate it to a specifically measured value, rather than a prediction, and then remeasure it locally. Just a thought, as I've got a way to go.

Edit: My understanding of the satellite data is that it was intended to provide a map at 20,000 ft above sea level, i.e. for air traffic, and that it doesn't necessarily provide highly accurate data at ground level. But I'll take a look at it again.

Edited By S K on 24/02/2023 17:33:50

I wanted a movable frame since I'd like to test it at a location where g is known to great accuracy. Otherwise, I can calculate g all day long and not have much clue as to whether it's anywhere near correct or not.

I constructed it in an "A" configuration to hopefully help stiffen it in the direction of swing. I haven't bolted the plastic to the aluminum profile yet (don't have enough T-nuts), but the fit is so tight that I don't think it would help anyway. I was thinking of adding braces connecting the sides of the A's, though, as the aluminum does "ring" a little if struck, and that can't be good. Obviously, it still needs a base quite badly, too.

That said, it's never going to be as rigid as optimal (optimal being a massive stone foundation wall in the basement of a remote castle), but the pendulum is fairly short and light, so with a few improvements I think it will do for now.

I don't believe the counter has the OCXO. I would expect it to be distinctly better than a plain Arduino, at least. But for the "g" measurement, I believe finding the center of gravity and the length measurements will inevitably be the limiting factors in accuracy.

Yes, it's very sensitive to the environment. My last test had lower noise for the period (6-ish us rms), just from less walking around it, and I think it should go lower still with care. I don't know what a "good" value for that is, though.

Edited By S K on 24/02/2023 15:55:29

Yes, I noted the internal pull-up resistor on the spec sheet, and the additional external one in their test schematic, too. I was happy enough with the ~20 ns fall time and didn't need both edges. Since adding another pull-up would likely worsen the fall time a little (while benefiting rise time), I left out the suggested external pull-up resistor.

And yes, placing the anvils on the pendulum was a lot easier than trying to fix knives to it!

Thanks.

Status and Next Steps

Generally speaking, I’m reasonably happy with the pendulum and its platform so far, and early performance measurements - despite some sketchy details in how it’s set up right now - seem good.

The pendulum’s support needs a base to add mass and rigidity. It’s just sitting free on its legs for now, and is not terribly stable or even properly leveled. A proper base, leveled properly, should make a good difference.

I need to polish the knife edges (they are just stock tool steel blanks). This should reduce friction further.

I need to add an adjustment weight to the pendulum and make an effort to equalize the periods. This would allow a comparison between Kater’s classic method and Repsold/Bessels’s formula. I then need to find the center of gravity of the pendulum (after adjustment) as well as measuring the distance between the anvils and between each anvil and the COG.

I only have an engineer’s rule to do distance measurements. Early on, microscopes were used to measure the knife (or anvil) positions. Instead, I have experimented with using macro-photography to do pixel-level interpolation. This is not easy, but I’ll discuss that later.

Finally, I'll calculate g at my location, or possibly someplace where g is known to high precision.

Questions or comments are appreciated.

Edited By S K on 23/02/2023 20:16:30

Electronics

I purchased a few Sharp GP1A57HRJ00F opto-interrupters. I measure about a 20ns fall time, but a far longer rise time. I’d recommend only using the fall time when measuring time periods.

For now, I’ve borrowed an Agilent 53230A frequency counter to do period measurements. This instrument uses a 10 MHz internal clock for timing, which is likely much better resolution than Kater was able to achieve, and so the time resolution of period measurements should not be the limiting factor in the measurement of g.

Preliminary Period Measurements

The pendulum currently lacks an adjustment weight that would be used (as in Kater’s pendulum) to equalize the periods when hung in both directions. I intend to add a weight and adjust the periods as best as I can, but for now, here are the periods for arbitrary swing amplitudes:

• Period for “normal” orientation (no adjustment weight): 1.528529 s.
• Period for “reverse” orientation (no adjustment weight): 1.449025 s.
• The noise in the period measurement (normal orientation): sigma = 8.9 us.
• Signal/noise ratio for a single period measurement = 172,000

I initially rested my instruments on the same bench as the pendulum, and measured noise over 10 times worse. Moving them (and their fans) off the bench reduced noise substantially! I believe there is more to be gained here with some effort.

Measuring Q Factor

As there is no restoring force applied, I could measure Q by the simple technique of counting the number of oscillations before the swing is reduced to 1/e and multiplying by 2*Pi. I placed an engineer’s rule behind the pendulum and used my phone’s video camera to estimate the amplitudes of the swings. With 60 Hz frame rates, this appeared OK for about 0.5mm resolution.

• Q (preliminary) = 18,500.

Edited By S K on 23/02/2023 20:08:56

The Pendulum’s Mount

A 3D printed PETG platform was printed to hold the legs of the stand and provide a (hopefully) flat surface for mounting the knives. A brass stage was cut to rest on top of the plastic platform. Initially, I had intended to implement a leveling apparatus between the brass stage and the plastic platform. However, the brass was found to be a little too thin to avoid slight bending under force, and plastic platform was found to be quite flat anyway, so in the end I mounted the two together. I estimate the deviation from flatness of the combination to be at or below 0.001” across the 6.5”platform.

On the brass stage, I mounted two V-blocks to hold the knives, which are 0.25” square tool-steel lathe blanks. It’s critical that these two knives have their edges exactly aligned with each other. Basic alignment was made by bolting a precision-ground rod in the two V-blocks before bolting the V-blocks to the brass stage. After removing the rod, I estimate that the deviation from parallelism between the two knives amounted to under 0.04 degrees deflection from horizontal, due to a very slight curve in the plastic platform that was transferred to the brass. This deflection means that the knives may bite an edge of one or both of the sapphire anvils rather than ride flat. I should be able to shim some of this out too, but the current result actually appears quite OK for now. If I redid this, I’d use thicker, stiffer brass and float it above the plastic, e.g. on 3 points, rather than bolting it to it.

Below is an image of the pendulum when it's mounted "upside-down" in the pivots.

Edited By S K on 23/02/2023 20:12:53

The Pendulum’s Frame

The frame for the pendulum is a free-standing “A-frame” rather than wall mounted so that it can be moved. The “A” leans in towards the direction of the pendulum’s motion to provide better rigidity in that direction. It’s made of 600mm long 2020 aluminum profile. This is smaller than ideal (Kater’s original was more like 2m long), but it’s proportioned for the ~1.5 lb. bob weight and 0.25” shaft diameter that I have. The frame is tied together by 3-D printed joinery made of PETG plastic.

I’m still deciding on a base material and configuration. As it lacks any at all at the moment, the structure is not very stable, but is OK for preliminary measurements.