Pendulum ‘Q’ value and measurement methods

[Joseph Noci 1Participant @josephnoci1]

[Joseph Noci 1Participant @josephnoci1]

There are a number of Pendulum threads running and I am doing my utmost to not catch the bug, but it seems I may succumb…

It is too tedious to go through paragraph in every post to find out – cannot just search for 'Q' in pendulum or clock posts…

What sort of Q values are being achieved by the various Pendulums worked on in these forums?

How are you measuring and calculating Q? There are a number or methods it would appear, and some appear contradictory.

What is the REAL value of Q in accessing a Pendulum's potential performance? There are many articles, some by real Gurus it would appear, who play down it's usefulness to some or other degree.

I understand the Quest for high Q's, but once a sound suspension method is found, drag is reduced to a sensible minimum, etc, what sort of Q value is usable/workable? Obviously aimed at a decent performance pendulum, perhaps not the Short…

Edited By Joseph Noci 1 on 08/08/2023 17:40:09

[S KParticipant @sk20060]

I measured a Q of about 18,500 in my genuinely-free pendulum (no restoring power input). The pendulum was about 24" in length, had about a 1 lb bob, and it was rocking on knife edges:

Gravity pendulum link

I measured it by counting the number of swings until the amplitude decayed to 1/e (to 36.8%) of its original value. I used a video camera to capture the amplitude of the swings, with an engineer's rule behind the bottom tip of shaft.

In a vacuum, Q will go up dramatically, since air resistance becomes a non-issue and essentially the only remaining loss of energy is in the hinge. The value of achieving high Q has been debated, but I have faith that obtaining high Q promotes high performance.

[Russell EberhardtParticipant @russelleberhardt48058]

Easy way to measure pendulum Q is to set it to swing freely and count the number of periods it takes for the amplitude to fall by 21%. Double it to get the Q. Expect a value of around 10,000 for a good clock pendulum.

Russell

Edited By Russell Eberhardt on 08/08/2023 18:05:17

[Michael GilliganParticipant @michaelgilligan61133]

I would be interested to know at what starting-angle of swing the pendulum is being checked.

It has long been common practice, in accurate clocks, to use very small angles of swing to avoid the effects of circular error … but the practical measurement of the delta reduction must then become increasingly difficult.

Mmm

MichaelG.

[Joseph Noci 1Participant @josephnoci1]

So, we have :

number of swings till amplitude = 36.8% of start

Number of swings till amplitude fell by 21%, and double the number

another source…Number of swings till amplitude is halved, time 4.53.

Where do these magic numbers come from?

Q actually = total energy / energy lost per swing, but that is difficult to measure…

[Russell EberhardtParticipant @russelleberhardt48058]

They are not magic numbers. They come from a bit of mathematics.

See for example **LINK**

Russell

[Michael GilliganParticipant @michaelgilligan61133]

Clever maths … but the resulting value for Q [ upper-right of p5] is woefully low.

MichaelG.

[SillyOldDufferModerator @sillyoldduffer]

My experimental pendulum reports 21399

It's calculated using the bandwidth definition because logging a pendulum with a microprocessor produces all the data needed to calculate bandwidth. Much easier that measuring decay and, I think, measuring impulsed Q is more useful than measuring free-swinging Q. Clock pendulums are impulsed, not free.

Bandwidth is calculated from the 29.3, 50,0 and 70.7 percentiles.

Easy in Python because numpy has a percentile function. Where tickArray is a list of periods:

h, l, r = np.percentile(tickArray, [70.7, 29.3, 50] )
b = h – l
q = r / b


Usefulness of Q? The balance wheel in a marine chronometer has Q of about 400, that of an ordinary well-made pendulum is said to be about 10000. Best vacuum enclosed pendula, up to about 100,000.

I see Q as a measure of the purity and stability of an oscillator, where high Q means it produces close to one frequency only. Real oscillators do not produce a single frequency, they wobble around it, and they are likely to drift.

I've noticed good long-term time-keeping can be got from a low Q pendulum – less than 5000. I believe it's because wobble errors tend to average out, and aren't obvious unless the clock has a high resolution display. As most pendulum clocks only display to the nearest second or minute, low Q may not matter because the error is invisible and doesn't accumulate. However, high Q becomes important whenever a clock must be high-precision or high-resolution.

I need a high Q pendulum because I'm chasing milliseconds and below, but what I'm doing is bonkers!

One of the other threads asked if Q varies. Normally Q is measured once by simple methods and assumed to a constant. My pendulum's Q isn't!

The data suggests the Q of my pendulum varies between about 15000 and 31000. The red line shows temperature and I think I see a relationship – after a lag, temperature causes Q to vary. Far from convincing though – I don't know what causes this. Hard to think why Q should repeatedly peak over a 100 hour period. Perhaps the moon did it!

Like as not other pendula do the same, because mechanical or environmental changes are likely to affect the purity of the signal. Anyone happy with their clock is advised not to look too closely! That way lies madness. Nature conspires in many ways to subtly alter the period of a pendulum despite the clockmaker's best efforts.

Dave

[duncan webster 1Participant @duncanwebster1]

Posted by Joseph Noci 1 on 08/08/2023 19:28:56:

So, we have :

number of swings till amplitude = 36.8% of start

Number of swings till amplitude fell by 21%, and double the number

another source…Number of swings till amplitude is halved, time 4.53.

Where do these magic numbers come from?

Q actually = total energy / energy lost per swing, but that is difficult to measure…

Probably something to do with exponential decay

[S KParticipant @sk20060]

Posted by duncan webster on 08/08/2023 20:44:58:

Posted by Joseph Noci 1 on 08/08/2023 19:28:56:

So, we have :

number of swings till amplitude = 36.8% of start

Count the swings to 36.8% … and then multiply that by 2*Pi. (Sorry, I left that out somehow.)

[Joseph Noci 1Participant @josephnoci1]

Posted by Russell Eberhardt on 08/08/2023 19:54:44:

They are not magic numbers. They come from a bit of mathematics.

See for example **LINK**

Russell

'Magic' was just a bit of a dig at the variation of methods.

Qilin Xue's maths appears sound but I would presume the other methods have some mathematical basis as well? Yet they would appear to give very different results?

The 'other' source I quoted comes from this link : LINK

[John HaineParticipant @johnhaine32865]

The Shortt free pendulum in vacuum IIRC had an estimated Q of ~100,000.

My tungsten bob pendulum has a measured Q around 24,000 or maybe more at lower amplitude. This measured by run-down tests over a period of hours.

Bateman's clock is about 12,000.

Classic paper by Bateman has a graph plotting accuracy against Q for a wide range of clocks from watches to atomic and showing inverse correlation over several orders of magnitude.

But Clock B only in the 4000 – 5000 range for complicated reasons. So Q is not the whole story.

There are many ways to measure and calculate Q but they are all consistent.

 

Edited By John Haine on 08/08/2023 22:27:39

[S KParticipant @sk20060]

All of the different swing-counting methods, e.g. "count swings until X% then multiply by constant Y" should yield the same results (I haven't checked, though).

It wasn't said, but I presume SOD's measurement was in a partial vacuum? I doubt that a pendulum's Q should change by 50% just due to temperature. I'd think Q certainly could change that much if the air pressure was changed, though. Barring that, I'd rather suspect that instantaneous measurements of Q done that way will be noisy. By comparison, the counting methods have averaging of many hundreds of swings built-in.

[Joseph Noci 1Participant @josephnoci1]

Thanks Chaps – more reading to do!

[Michael GilliganParticipant @michaelgilligan61133]

Nothing new here: **LINK**

https://www.statisticshowto.com/calculus-definitions/damped-sine-wave/

Just standard statistical stuff, nicely presented.

MichaelG.

[SillyOldDufferModerator @sillyoldduffer]

Posted by S K on 08/08/2023 22:33:50:

…

It wasn't said, but I presume SOD's measurement was in a partial vacuum? I doubt that a pendulum's Q should change by 50% just due to temperature. I'd think Q certainly could change that much if the air pressure was changed, though. Barring that, I'd rather suspect that instantaneous measurements of Q done that way will be noisy. By comparison, the counting methods have averaging of many hundreds of swings built-in.

Ah, no.

• My pendulum is running in air at the moment because I haven't built the vacuum plumbing yet
• I'm not measuring instantaneous Q. I have a dataset of 2,314,308 period records (roughly 600 hours at 0.93s per beat) Q is calculated in 1 hour slices, so each Q comes from nearly 4000 samples. I submit this is substantially better than the counting method.

Is the variation pressure related? I don't think so, here's the graph:

Date-times that match the Hour Numbers above:

clockStart = 2023-07-15 20:30:03 (Hour zero)
ClockEnd = 2023-08-09 20:43:29.861326 (1691610209.0)
ActualEnd = 2023-08-09 20:43:01.319747 (1691610181.0)

ClockRun = 25 days, 0:13:26.861326 (2160806.861326s)
ActualRun = 25 days, 0:12:58.319747 (2160778.319747s) (600.0 hours)

(ClockEnd is finish time as measured by my clock, ActualEnd is the same finish according to Network Time Protocol. On my set-up NTP is no worse than about 100mS different from Atomic Time, and is normally better, roughly 25mS

The Counting and Bandwidth methods produce slightly different results. Doesn't matter, I think, because Q-factor is more indicative than absolute. Q gives a good idea of pendulum quality, but is far from the whole story. Allan Variation is what's needed for that, but despite several attempts I still don't understand it! If anyone can explain the Wikipedia Article to me I shall be eternally grateful.

Dave

[Michael GilliganParticipant @michaelgilligan61133]

This is not a criticism, Dave … far from it !

But something is nagging me: By [ quite understandably] scaling your pressure graph from 970, are you perhaps exaggerating its significance ?

… what would it look like if you scaled it from zero ?

MichaelG.

[S KParticipant @sk20060]

SOD, I know you believe you are measuring Q, and you are getting "numbers," but I don't believe you are actually measuring Q. The old "garbage in, garbage out" problem, to my eye. If your data was Gaussian and well behaved, and if the obtained value of Q was stable, maybe, but at this point none of that seems evident.

Also, I am not sure you are using thousands of samples. Sure, you are collecting thousands, but per Q measurement you are throwing nearly all of them out after selecting only a few (I don't know the details, however, so I could be wrong on this point).

In addition, you are subtracting two numbers that are very close together from each other, getting a very small number (i.e., 6 or so orders of magnitude smaller), and then dividing that very small number into a comparatively large one again, tempting the gods of mathematical fate. In this scenario, minute deviations can cause huge impacts on the end result, as it seems you are seeing.

Try using the decay method to check. It can't be that hard, others have done it, and the decay method is much more intuitively related to the loss of energy per swing anyway. And also, the value of Q obtained this way should be quite stable from trial to trial (as I believe one would expect from a macroscopic pendulum of this sort).

 

Edited By S K on 09/08/2023 23:02:19

[duncan webster 1Participant @duncanwebster1]

I'd expect Q to change with atmospheric pressure and perhaps humidity. Intuitively I'd go for run down to measure it, but intuition based on little actual experience can be dangerous.

[S KParticipant @sk20060]

Duncan, yes, higher pressure and (probably) humidity should reduce Q.

But if I'm reading the graph right, a 3.5% atmospheric pressure change is plotted against a 100% change in Q.

[John HaineParticipant @johnhaine32865]

I think S K's points about measurement error are good. If your amplitude measurement is based on a pulse length which is the small difference between two numbers, then every measurement has an error. If you then difference successive amplitude measurements then again you get errors stacking up. When I has estimating the Q of my Arduinome IIRC I got reasonably consistent results looking at decay over the 60s impulse cycle where one could at least average 30 numbers – Q ~12,000 I think.

[Joseph Noci 1Participant @josephnoci1]

I am very green on this so please forgive questions that may seem very simple to those skilled in the art…

looking at decay over the 60s impulse cycle

What does that mean, and how do you do it?

My pendulum has a 3.5kg bob, inside of which is an XYZ accelerometer , and it has a slotted opto detector at BDC.

Ignoring the magnetic impulse mechanism fitted, for the moment, how do I practically measure Q with this setup?

I hope to have a pendulum swing of 2 degrees ( 1deg either side) . Can I measure Q by setting the pendulum swinging manually, and monitor the pulse and pendulum period say on a 'scope?

I have been playing and using SK's method I get Q's of 18000 to 20000 and I do not believe it….

A gentle help please!

[duncan webster 1Participant @duncanwebster1]

You could use the slotted opto, as amplitude decays the ratio of interrupted to open increases, then a bit of dimple maths, but I'll have to fire up the PC, trying to type it on a phone will drive me potty.

[John HaineParticipant @johnhaine32865]

Posted by Joseph Noci 1 on 10/08/2023 11:44:45:

I am very green on this so please forgive questions that may seem very simple to those skilled in the art…

"looking at decay over the 60s impulse cycle"

What does that mean, and how do you do it?

The pendulum is impulsed once every minute by a gravity arm and the amplitude slowly decays until the next impulse (though not by very much). In equilibrium the max and min amplitude over the impulse cycle are constant from (impulse) cycle to cycle. The amplitude is estimated for each pendulum cycle by an opto and a picPET, deriving the amplitude from the pulse length. The decay of the maximum over 30 cycles closely follows a reverse-exponential law: An/Ao = exp(-n*pi/Q) where An is the amplitude of the n'th cycle and Ao that of the first. Just invert the expression to get Q.

My pendulum has a 3.5kg bob, inside of which is an XYZ accelerometer , and it has a slotted opto detector at BDC.

Excellent that you have an accelerometer! It will be very interesting to see the results. How wide is the slot (or vane)? A simple and reasonably accurate way to calculate amplitude as long as it is fairly large compared to the width and on-centre is just to compute the velocity and derive the amplitude from that and the period.

Ignoring the magnetic impulse mechanism fitted, for the moment, how do I practically measure Q with this setup?

What I do is to use the impulsing to get the amplitude up to a decent value a little larger than your target, and switch it off, then just observe the amplitude as it decays and apply the formula above.

I hope to have a pendulum swing of 2 degrees ( 1deg either side) . Can I measure Q by setting the pendulum swinging manually, and monitor the pulse and pendulum period say on a 'scope?

You could use a 'scope but I know you're a time-nut too so something like a picPET would be better!

I have been playing and using SK's method I get Q's of 18000 to 20000 and I do not believe it….

Re-calculate using the above formula, I am pretty certain it is right (and a mathematician friend agrees).

A gentle help please!

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