JoNo’s Pendulum

[Joseph Noci 1Participant @josephnoci1]

I have re-designed almost all the electronics on the pendulum. The new ideas implement better op-amp voltage offset and temp drift control and I have implemented a linear voltage controlled amplifier for the pendulum sinusoidal drive voltage amplitude control- this was logarithmic in the previous design and was very sensitive. Also, the 90deg drive signal phase shifter is now controlled via a 24bit DAC, from the Nucleo processor. The phase shifter gives plus and minus 5deg shift around 90deg with a -3v to +3v DAC control voltage.

The amplitude control loop works well – it locks quickly and appears very stable

I will post circuits etc later , for anyone interested.

I am trying to determine the change in period wrt the change in phase at the moment, to see if I can derive a transfer function I can use in the final software control loop. This means setting the pendulum for a 'stable' drift rate and logging the period , and then changing the phase in 0.5deg steps while logging. From that I should be able to derive the change in period at phase change, and how that works through the system for some 10s of minutes after the change.

This is interesting, since if the pendulum period is increased, for example, the amplitude had to have changed as well, and the amplitude control loop will try to bring that back into closed loop control. So, maybe the phase change will change the period, for a short while, till the amplitude loop stabilises again. Then removing the phase change repeats the process in the opposite direction – this is how it simulates anyway, so I think the phase change method of control will be an application of phase change for a short period, seconds, etc, just enough to affect the change in period, and then revert back to 90deg…

Lots to learn down this road.

I have a 'theory' question please…

To mechanically adjust the pendulum period I have a nut under the bob that raises/lowers the bob, for coarse adjustment.

At the top of the pendulum rod I have two adjuster nuts, one small (in yellow), another larger (in red), just below the pivot. These give very fine rate adjustment.

As expected, if I lower the bob, the rate increases. However, if I screw the large adjuster nut DOWN (away from pivot, towards bob), the rate DECREASES, and UP INCREASES the rate. Can someone defend this mathematically??

Edited By Joseph Noci 1 on 30/09/2023 20:38:22

[Joseph Noci 1Participant @josephnoci1]

A quick plot of Phase angle change and effect:

X axis is elapsed time in 100ms periods. Y axis is pendulum period delta from GPSDO 0.5Hz reference.

The left rising slope: Commanded phase angle = 86deg.

duration is 400seconds, with a pendulum period change of 5769us over said 400sec, giving a pendulum period increase of 0.005769sec in 400sec

The right falling slope:: Commanded phase angle = 93deg.

duration is 700seconds, with a pendulum period change of 5128us over said 700sec, giving a pendulum period decrease of 0.005128 sec in 700sec.

I believe this may just work…

[S KParticipant @sk20060]

First, I think you mean that the period increases when you lower the bob, not the rate.

If you add mass right at the pivot, there will be no impact on the period at all. If you lower the adjustment nut, the period will decrease. Think of that mass as detracting from the mass of the bob. If you add mass below the bob, the period will finally increase.

That's assuming that the rod is very light and the bob is very heavy, meaning that the center of mass and the radius of oscillation are both inside the bob.

In the other extreme, imagine a pendulum consisting only of a heavy rod (no bob). For this, the radius of oscillation is 2/3 down from the pivot. Add weight above that, and the period will decrease. Add below that, and it will increase.

Edited By S K on 30/09/2023 22:58:30

[Joseph Noci 1Participant @josephnoci1]

Posted by S K on 30/09/2023 22:46:03:

First, I think you mean that the period increases when you lower the bob, not the rate.

If you add mass right at the pivot, there will be no impact on the period at all. If you lower the adjustment nut, the …………

Add weight above that, and the period will decrease. Add below that, and it will increase.

Apologies, thanks for correcting me – period is what I meant, and thanks for the explanation

Joe

[John HaineParticipant @johnhaine32865]

Standard expression for effect of phase error on fractional change in rate is

Tan(phase error)/2Q

Which is one reason why high Q is desirable.

[SillyOldDufferModerator @sillyoldduffer]

Posted by Joseph Noci 1 on 06/09/2023 09:29:49:

…Pressure, on the other hand, seems to dominate.

The curves below:

…

seems to show that the swing is increased as the pressure increases which 'sounds' correct – the bob is slowed as pressure rises so the swing reduces, and the system gain then increases to get the swing back up again. It makes sense, but I am not sure.

…

Would have commented at the time except I missed this interesting post. I'm fretting about barometric pressure because it's almost certain my approach to compensating for pressure change is wrong. Long story and I'm still working on it, but I was educated by this description found on physics.stackexchange:

• Atmospheric pressure : The effect of the surrounding air on a moving pendulum is complex and requires fluid mechanics to calculate precisely, but for most purposes its influence on the period can be accounted for by three effects.

By Archimedes' principle the effective weight of the bob is reduced by the buoyancy of the air it displaces, while the mass (inertia) remains the same, reducing the pendulum's acceleration during its swing and increasing the period. This depends on the air pressure and the density of the pendulum, but not its shape.

The pendulum carries an amount of air with it as it swings, and the mass of this air increases the inertia of the pendulum, again reducing the acceleration and increasing the period. This depends on both its density and shape.

Viscous air resistance slows the pendulum's velocity. This has a negligible effect on the period, but dissipates energy, reducing the amplitude. This reduces the pendulum's Q factor, requiring a stronger drive force from the clock's mechanism to keep it moving, which causes increased disturbance to the period.

I'd assumed a simple linear relationship when it's complicated and non-linear! No wonder my clock doesn't work as intended.

I haven't quite given up on compensating mathematically for changing air pressure, but running the pendulum in a vacuum is the easiest way of fixing the problem. My mind goes blank when I tackle the maths, and I'm already struggling with a paper John Haine sent me!

Dave

Dave

[Joseph Noci 1Participant @josephnoci1]

Posted by John Haine on 01/10/2023 08:31:53:

Standard expression for effect of phase error on fractional change in rate is

Tan(phase error)/2Q

Which is one reason why high Q is desirable.

John, what phase error do you mean? Or which phase(s) are used to define this error term?

[duncan webster 1Participant @duncanwebster1]

SK you're going to have to explain that, Joe isn't adding mass, he's moving an existing mass. I'd expect moving the little nut down and leaving everything else alone to effectively lengthen the pendulum and so increase period

[John HaineParticipant @johnhaine32865]

Posted by Joseph Noci 1 on 01/10/2023 14:02:22:

Posted by John Haine on 01/10/2023 08:31:53:

Standard expression for effect of phase error on fractional change in rate is

Tan(phase error)/2Q

Which is one reason why high Q is desirable.

John, what phase error do you mean? Or which phase(s) are used to define this error term?

The phase difference between the impulses and BDC of the pendulum. In your case with a sinusoidal drive the maximum drive torque is ideally at BDC.

[John HaineParticipant @johnhaine32865]

Posted by duncan webster on 01/10/2023 16:17:51:

SK you're going to have to explain that, Joe isn't adding mass, he's moving an existing mass. I'd expect moving the little nut down and leaving everything else alone to effectively lengthen the pendulum and so increase period

Any majss added above the CG of the pendulum moves the CG up and effectively shortens it. For a given added mass its effect is maximum half way down the road, minimum at the bottom and top.

[duncan webster 1Participant @duncanwebster1]

But he isn't adding mass, he's moving existing mass down (unless I've completely misunderstood what Joe wrote

However, if I screw the large adjuster nut DOWN (away from pivot, towards bob), so the cg goes down. Not like adding pennies to Big Ben. if m is the little mass and its distance is d, M is bob mass and its distance is D,

distance of cg from hinge = (m*d +M*D)/(m+M) which increases as d increases

[S KParticipant @sk20060]

Considering a simple pendulum (weightless rod, point-mass bob): Sure, if you are not adding mass but merely lowering an adjustment weight, the CG shifts down. But the CG isn't the main point, the radius of oscillation is. That's the position that the point-mass bob would be if you converted a compound pendulum (bob plus adjustment weight) to an ideal pendulum with the same period.

Slide the adjustment weight all the way to the top, and that weight becomes irrelevant to the period since it's not even swinging. Slide it all the way down to the bob, and it's also irrelevant! Why? Because now you just have a heavier bob, and the period isn't dependent on the mass of the bob! (T=2*Pi*sqrt(L/g) – the mass m of the bob is not in the equation.) The CG is obviously quite different in those two cases, but the result is the same – no change in period!

Slide it to somewhere near the middle, though, and now you have distinctly changed both the CG and the radius of oscillation. In that case, the compound pendulum is like an ideal pendulum with the bob higher up than before, hence a reduction in the period (the pendulum swings faster). This effect is at a maximum when the adjustment weight is at the middle of the rod. You have to put the adjustment weight below the bob to increase the period.

It becomes a bit more complicated when the rod is heavy and the bob is light in comparison. Then, the radius of oscillation is already above the bob, so lowering an adjustment weight below that point (but still above the bob) can increase the period. That result came up in my very light "gravity pendulum," but wouldn't often be the case for a more typical pendulum (carbon fiber rod, very massive bob that's extended in space) that folks are making here.

Edited By S K on 01/10/2023 19:29:01

[duncan webster 1Participant @duncanwebster1]

well blow me down! I was so troubled by this that I had to do some sums. I took a pendulum with a bob 10 units in mass suspended 10 units from a hinge and then slid a 1 unit mass down the rod in 1 unit increments. Lo and behold the Equivalent Length starts at 10, goes down to 9.76 at 5 units, then back up to 10. the distance to the cg starts at 9.09 and goes up steadily to 10

 

If anyone wants to repeat the exercise, according to wikipedia the Equivalent Length is (m*d^2+M*D^2)/(m*d+M*D), using the same nomenclature as before. You're never too old to learn something! If it wasn't such a pain to import an image I'd paste it in

Edited By duncan webster on 01/10/2023 21:55:16

[Joseph Noci 1Participant @josephnoci1]

My pendulum exhibits an interesting rod 'vibration' mode..

I suspect this is inherent in all pendulums but passes unobserved due to the way the period is measured by most folk.

Since my pendulum is amplitude controlled by measuring the PP swing sinus voltage, the quality of that measurement is key to generating a clean stimulation sinusoid. The PP measurement is done by means of a 2 op-amp Peak detector ( typical 3 diode/op amp peak detector). My PP swing logging shows noise, which I believe stems from the peak detector – the peak detector works by 'topping' up a storage capacitor to the peak value of the sinus, but the opamp that generates this top-up voltage operates open-loop til top-up time, and the top-up interval seems to vary, which means the peak voltage on the storage cap varies, which means my drive voltage to the pendulum coil varies… Not much, 8 to 14mv, which equates to 14 millidegrees swing variation max. I am still unsure how to equate 14 millidegrees variation to a ppm value in pendulum stability…

However..

In pursuing a cleaner PP measurement method, I notice a regular modulation on the logged PP swing sinus – at first I thought this was the Peak detector somehow feeding through – I replaced the peak detector output with an equivalent fixed voltage. The pendulum was now not amplitude controlled, but was driven, with no peak detector artifacts able to affect it. Made no change.

I then disconnected the sinus drive to the pendulum coil, and let is run down to 10% swing. Made no difference to the modulation phenomena – did a Q calc to verify the pendulum is still 'OK' – Q = 12900, so OK.

Then I reconnected every thing up, did a run and logged it –

This shows a modulation depth of around 10mv – 10milligrees at a 50 second period.

I then moved/fiddled with the knife pivot location point, ie, moved the knife forward about 0.2mm, so it was 'riding' up the inner ball race support surface. This meant the knife was twisted in its circular support pivot, and would add some odd motion to the swing.

I then logged data again – I did observe a new modulation on the Swing sinus, but at a 2sec period, ie, same as the pendulum period – I could also see the bob was swinging in an ellipse, rather than a straight line. However, the 50sec modulation period remained unchanged, so it was not induced by the knife or pivot or its friction, etc.

I then added a weight to the pendulum rod – an AA penlight cell fitted with masking tape.. – at 40% down from pivot, and logged data –

This showed the same modulation depth approx, but the period was now 35 seconds.

I then added 2 more weights ( AA cells) one at 20% down, one at 65% down.

The period changed to 25 seconds.

So – the rod somehow has a very long period vibration mode, or a High frequency vibration mode, which influences the pendulum period in a very cyclic manner. I suspect everyone using light weight 'stiff' rods has similar modes, but do not see them due the to measurement methods.

This shows the three logs, no weight, one weight 3 weights.

Expanded view of the bottom peaks of the sinus swing, shows the 3 varying modulation frequencies

No weight

1 Weight

 

The carbon fibre tube is a no-no – maybe a solid carbon fibre rod will improve matters, but that prevents fitting wires internally to my sensors in the rod and bob. And solid carbon rods temp coef is a lot poorer than a tube, so I would be better off using Invar…

And I am pretty sure the house is not vibrating at those frequencies…

The issue is evaluating how much of an effect this modulation has on the pendulum period – it is cyclic, 10 millidegrees, which can be validated in change of period, but the cyclic nature undoes what it causes on a regular basis. Not sure if I am chasing ghosts here.

 

 

Edited By Joseph Noci 1 on 03/10/2023 08:49:25

[Joseph Noci 1Participant @josephnoci1]

I have been digging in internet to try find some info on this phenomena and come up dry. Searching 'pendulum rod' vibration brings up people who call the pendulum swing a vibration, and the rod vibrates at pendulum period…Even Alan Cromer's 'many oscillations of a rigid rod' does not help…

What is also interesting is that the modulation frequency as described has seemingly no relationship to the rod's 'Twang' ( there's that word again) frequency, which is in the many 10s of Hz region.

Edited By Joseph Noci 1 on 03/10/2023 10:47:20

[Michael GilliganParticipant @michaelgilligan61133]

Posted by Joseph Noci 1 on 03/10/2023 08:42:32:

My pendulum exhibits an interesting rod 'vibration' mode..

[…]

So – the rod somehow has a very long period vibration mode, or a High frequency vibration mode, which influences the pendulum period in a very cyclic manner. […]

.

That is VERY interesting Jo, but I honestly can’t imagine any single vibration mode of your pendulum which could have that [those] period[s] … I therefore suspect that you are seeing a ‘beat frequency’ between two other modes.

How the devil to identify them is way beyond my ability.

MichaelG.

[duncan webster 1Participant @duncanwebster1]

This is not a serious suggestion, but I've just re-read Hooker's autobiography where he mentions shedding of Karman vortices as a cause of vibration, so I thought I'd throw it in to add to the confusion.

[Joseph Noci 1Participant @josephnoci1]

Not having a copy of Hooker's autobiography, I entered 'Hooker's autobiography' in the google search line.

I know I am plagued by a vibrating rod, but the first few search results were…interesting…

Anyway having got past that, I dug further to see how this might assist – I would assume it is not what is happening though. Since the Karmen vortex shedding frequency tends to be similar to the natural frequency or resonant frequency of a structure/wire/cylinder, I would expect to see a vibration frequency similar to that emitted by the rod after twanging it – not 25 to 50second,

I must say I am at a complete loss here. I can find no refence to this. Also I am not sure if this a potentially limiting effect, or literally just noise in the system.

[duncan webster 1Participant @duncanwebster1]

Sorry, I meant this Hooker, expert on superchargers and jet engines. Well worth a read, Title Not Much of an Engineer available via Amazon in both paperback and Kindle. The paperback is significantly cheaper.

I did say it wasn't a serious suggestion!

[SillyOldDufferModerator @sillyoldduffer]

I've noticed 'twang' too. Likely it occurs in all pendula, but isn't noticed unless the apparatus can measure period at high-resolution.

I suspect many pendulum noise phenomena tend to cancel out over many beat. Noise only starts to become an issue when a clock keeps better than seconds, and is only obvious when microseconds are clocked.

I think the most likely cause of 'twang' is the rod flexing. An impulse applied at the top travels down a long lever to move the bob, which has inertia. The rod must bend, and I'd expect it to oscillate. Moving through air must also vibrate it slightly. Not much, but detectably.

If the rod was a violin string, what frequencies would it vibrate at? (Violin notes are far from pure – they twang too.)

Dave

[Michael GilliganParticipant @michaelgilligan61133]

Dave,

The two lowest frequencies of a pendulum rod must surely [?] be its first bending mode and its first torsional mode … everything else, I posit, will have a higher frequency.

The magnitude and frequency of both of these will depend upon the construction details of Jo’s pendulum [and they should be simple for him to measure] … but I would expect both to be many octaves above the frequencies he is observing.

… hence my clutching at the ‘beat frequency’ straw.

MichaelG.

Edited By Michael Gilligan on 03/10/2023 19:41:38

[Michael GilliganParticipant @michaelgilligan61133]

Posted by SillyOldDuffer on 03/10/2023 18:46:34:

[…]

If the rod was a violin string, what frequencies would it vibrate at? (Violin notes are far from pure – they twang too.)

Dave

.

Oh dear … wasted my time again

**LINK**

https://www.model-engineer.co.uk/forums/postings.asp?th=186038&p=6

20/04/2023 22:57:47

.

MichaelG.

[S KParticipant @sk20060]

Interesting stuff. One point: "And solid carbon rods temp coef is a lot poorer than a tube…"

I'm wondering why this should be? Thanks.

[Michael GilliganParticipant @michaelgilligan61133]

Many, many variables in carbon-fibre composites !!

This is probably the most pertinent paper I have seen: **LINK**

https://opg.optica.org/oe/fulltext.cfm?uri=oe-26-1-531&id=380577

The PDF is a free download

MichaelG.

Edited By Michael Gilligan on 03/10/2023 20:54:00

[Joseph Noci 1Participant @josephnoci1]

Posted by duncan webster on 03/10/2023 18:33:42:

Sorry, I meant this Hooker, expert on superchargers and jet engines.

Duncan, yes, I did find a web copy of that during my search and did read, hence my comment regarding its applicability. There is just so much interesting stuff out there, I get waylaid and the hours just pass…

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