JoNo’s Pendulum

[Joseph Noci 1Participant @josephnoci1]

The Pendulum so far –

I opted for a knife edge style pivot – 6x6mm cobalt section 100mm long as the knife, riding on the inside race of a ball bearing. The bearing was loctited into its loose fit housing, with a ground shaft fitted through both bearings to locate and align the races. once cured, loctite was added to the balls in the races, the inner race rotated to spread the loctite, and left to cure with the shaft aligning the races.

The head assembly with knife pivot can pivot laterally on hard steel pins in two V block, on the top support assy –

The frame is from sq tubing, with the base of 50x50mm tube, filled with lead shot, and the space between with a chunk of St-steel – 210x110x50mm (!) – base is about 15kg.

The bob is made in three parts that screw together so I can fit electronics inside – accelerometer, temp sensor, etc

The Pendulum rod is a carbon fibre tube, 7mm OD, 6mm ID. The wires to the bob run inside the tube and out the top at the pivot point so cable torsion is minimised. The wires come out the tube via thin (0.05mm thick) flexi PCB's.

My intention is to provide a sinusoidal drive to the pendulum coil, not impulse. This drive to be phase locked to the pendulum motion, with the amplitude of the drive tuned through a Kalman filter to provide sufficient amplitude to just keep the pendulum amplitude the same, and thereby hopefully the period constant. The idea being not really to keep the amplitude the same but rather the total mechanical energy must be kept to a constant value. I am not driving for a specific period, but for constant , fixed energy. If the period is not exactly one second, I will gear this in software to drive an external clock at 'true' 1 second.

The drive to the pendulum is at the pivot end, not the bob end. It uses the drive coil from a small hard disc drive, and is sandwiched between the very powerful drive magnets.

This is the head assy and the magnets in there iron concentrators can be seen here..

Below is the pendulum tube top end, with the two screw adjustment weights – the larger weight is 15 grams, the smaller is 6 grams. The hole to the right of the small adjuster is where the knife will fit.

The magnets in place with the sandwiched coil, and the adjusters under seen here below

(edit- bugger this sofware! have to split the post again to many words….!!)

[Joseph Noci 1Participant @josephnoci1]

[Joseph Noci 1Participant @josephnoci1]

The pendulum achieves a +- 1deg swing ( about 30mm P to P) after 9 applications of 12mA through the coil, applied one way with polarity for the swing direction. This drive will be sinusoidal in the final solution.

So far the accelerometer is useless. The rates are to low for simple MEMS gravity type Acceleros. So I designed a capacitive angle sensor that gives 1.8V P to P for a +- 1 deg pendulum motion. This is very linear, and uses synchronous detection of ac signals to eliminate DC offsets and drift, so is very stable with temp and voltage variation.

Top left is the sensor drive plate, top right the pickup plate, bottom left the shield vane, and bottom right fitted on the head,

All the associated electronics…

Scope output of the sensor with a +- 5mm swing..Scope trace shows 40mv PP, but the probe is X10, so actually 400mv PP. gives 2V PP for a 35mm PP swing.

I set the pendulum swinging 25mm total swing, with a scope showing 1380mv PP. I let it run till the scope show 36% of that – around 50mv. It ran for 2hours 30min, with a period of 2sec ( see scope – this was some minutes after the end of the run..) . According to SK's Q formulae that gives 4500 swings over 2H30m, times 2 times pi which is a Q of 28,0000.

Sorry, not sure I believe that formula…

[S KParticipant @sk20060]

WOW!

[John HaineParticipant @johnhaine32865]

Very interesting approach Joe!

Sinusoidal drive was used in a couple of clocks by Ned Bigelow in the US. He used a moving magnet design with a large number of turns on his coils. This gave a high transduction coefficient "K" which allowed him to connect the pendulum as the resonator in a simple oscillator circuit – in effect he had a resonant circuit with the Q of the pendulum. IIRC "K" is the volts/m/s as well as newtons/ampere – same value different units. I looked at using a similar approach with my design but K is very small so the impedance would be swamped by the circuit parasitics. Do you have any measurements for your drive arrangement? I did consider using sinusoidal drive in the way you are proposing and still might, so looking forward to seeing your results.

On the suspension, the arrangement you have for the trunnion bearing makes it hard for the pendulum to hang truly vertical "back and forth" as there will be some friction between the bar and vees (yes, I know I use a similar arrangement!). IIRC the maximum hanging angle is something like arcsin(coefficient of friction), it's in Matthys' book. Doug Bateman simply has a flat surface on the "brackets" so the pendulum can roll to verticality. Or one could use ball bearings to reduce friction but also constrain the pendulum back-and-forth.

[Michael GilliganParticipant @michaelgilligan61133]

Predictably wonderful, Joe

… I will be following this with great interest

MichaelG.

[SillyOldDufferModerator @sillyoldduffer]

This sort of post does nothing for my inferiority complex! Loads of good ideas, extra well-made, and Joe is another one who must work at least twenty times faster than me.

Disappointing results from the accelerometer were predictable, but good to see it tried, especially in a pendulum design that allows that sort of experimentation. Though I like sensors inside the bob, I worry about wires flapping about inside the rod and adding friction where they exit at the top. (Causing much more friction than the knife-edge?)

May be worth adding a humidity sensor. Although almost completely insensitive to temperature change, my samples of solid Carbon Fibre rod were humidity sensitive. I believe humidity acts on the matrix and alters its elasticity. If running the pendulum for a long time shows anomalies, humidity might be the cause. (In the UK humidity varies a lot: I've no idea how much it changes on a Namibian sea-shore!)

Can't wait to see how it performs.

Dave

PS Poor Joe. After resisting the pendulum bug for ages he's caught it badly. Worst case of pendulitus I've ever seen. Welcome to the dark-side…

[Joseph Noci 1Participant @josephnoci1]

Posted by John Haine on 16/08/2023 07:49:50:

Very interesting approach Joe!

….. Do you have any measurements for your drive arrangement?

On the suspension, the arrangement you have for the trunnion bearing makes it hard for the pendulum to hang truly vertical "back and forth" as there will be some friction between the bar and vees (yes, I know I use a similar arrangement!). IIRC the maximum hanging angle is something like arcsin(coefficient of friction), it's in Matthys' book.

No measurements yet, other than the Q value I stated, which I do not believe at all I fear, but the numbers are what they are, so the formula must have a problem? maybe a times 2 or a times pi to many..?

The trunnion – I followed the description in Matthys' book – the included angle is 115 deg, not his 120deg that he speaks of ( I had a cutter of 115deg..and polished the V and the hard pins. At the moment there is also a very slippery thin moly lube in the V as well. I made an impossibly finicky setup to measure the hysteresis in front rear swing – a thin wire ( cats whisker) mounted on a micrometer rail, making (just) contact to the bob. Pull the bob away 5mm and return it gently, using a piece of cotton yarn , till the yarn goes slack, and then advance the micrometer till contact is made again. Do it 10 times, and then do it from the otherside – I get a spread of 'estimated' 0.005/0.008mm – difficult to read on the micrometer. That is very small and a very small angular delta therefore. The pendulum swing across the trunnion dies out to almost nothing within 18 seconds when started with a +-10mm P to P swing.

[Joseph Noci 1Participant @josephnoci1]

Posted by SillyOldDuffer on 16/08/2023 10:33:34:

Disappointing results from the accelerometer were predictable, but good to see it tried, especially in a pendulum design that allows that sort of experimentation. Though I like sensors inside the bob, I worry about wires flapping about inside the rod and adding friction where they exit at the top. (Causing much more friction than the knife-edge?)

May be worth adding a humidity sensor. Although almost completely insensitive to temperature change, my samples of solid Carbon Fibre rod were humidity sensitive. I believe humidity acts on the matrix and alters its elasticity. If running the pendulum for a long time shows anomalies, humidity might be the cause. (In the UK humidity varies a lot: I've no idea how much it changes on a Namibian sea-shore!)

Can't wait to see how it performs.

Dave

PS Poor Joe. After resisting the pendulum bug for ages he's caught it badly. Worst case of pendulitus I've ever seen. Welcome to the dark-side…

The accelrometer does give some data, but it can only be used in tilt mode, combining two axes as sine/cos and computing tilt, but there are not enough bits resolution for usel work.

I have obtained a 0.5G device, with 16bits resolution ( the other was 12 bits over 2G), so it may be better. But the Angle sensor works just fine.

No wires flapping in the rod – the rates are so low…And no friction of wires at exit there is a flexi pcb , very thin, at a 35mm radius bend, so adds no friction and no spring.

There is a temp sensor in the BOB and two thermisters in the carbon tube, one 80mm from bottom, the other midway up – all to the arduino in the BOB and then only 3 wires aout Pwr, Gnd and serial data @ 57Kbaud

At the pendulum head, where all the sensor electronics sits, is a Nucleo @ 140MHz, with ambient temp Pressure, sensor, and humidity sensor.

Humidity here is around 85% in winter @ 12-14deg C, and around 55-65% in Summer @ 18-22deg C.

Pressure here is typically 1024mb, and goes down to 1010mb at times…

[Joseph Noci 1Participant @josephnoci1]

Please someone guide me with this darned Q issue – I know Matthys and others don't place all the emphasis on Q alone, but is is a measure used by many to company baseline pendulum numbers.

Shortt pendulum was around 22000 in air I think? and mine is 28000??? Yeah, sure…

[John HaineParticipant @johnhaine32865]

An/A0 = exp(-n*pi/Q), where n is number of periods. So if ln(An/A0) = -1, then Q = n*pi.

So An/Ao = exp(-1) = 0.6321. Count cycles until amplitude is 63.2%, multiply by pi.

[Joseph Noci 1Participant @josephnoci1]

Thanks John.

This is what worries me…SK indicated a formula, also from sound maths, that is essentially, count beats till amplitude reaches 36%, then multiply by 2pi.

Till 36% is a lot more beats than till 63%, and times pi and then still times 2….

Your way seems to give me a Q of around 6000 – now that would be really disappointing.

[AlanParticipant @alan14594]

Philip Woodward, in his book.. My Own Right Time, page 17 gives…

Q = 2Pi x (total vibrational energy / energy lost per period)

which he says can be proved that Q = Pi /(loge 2) x the number of periods (double swings) the pendulum takes in decaying to half amplitude.

pi / loge 2 = 4.53.

His pendulum came out with Q = 14,000 (approx)

Edit… Loge is exponential logarithm

Alan

Edited By Alan on 16/08/2023 13:21:13

[S KParticipant @sk20060]

I don't know where I got that formula from, but apparently it is wrong. And that would mean that my measured Q just got down-graded too. 😒

Everyone seems to be using a different formula or different method. This needs to be straightened out.

[Joseph Noci 1Participant @josephnoci1]

Posted by S K on 16/08/2023 13:39:38:

I don't know where I got that formula from, but apparently it is wrong. And that would mean that my measured Q just got down-graded too. 😒

Everyone seems to be using a different formula or different method. This needs to be straightened out.

That's what I said 10 days ago…

Wait for amplitude to:

Drop by 21%, then times 2 =Q

Drop to 36%, then x 2 then x pi = Q

Drop til 50%, then x 4.53.

Drop till 63%, then times pi

The last 4 runs I did on my pendulum I captured swing counts at every 2% drop.

Using those numbers with the above wildly differing formula gives me Q's anywhere from 6000 to 28000.

At the moment I trust John Haines info , even if the result entices me to stop this nonsense before I have even started..

And the loose use of terms is also messy – Swings, periods, double swings, beats…..

[Michael GilliganParticipant @michaelgilligan61133]

By its very nature, Wikipedia cannot be considered authoritative … but this looks a very good place to start: **LINK**

https://en.wikipedia.org/wiki/Pendulum#Q_factor

Reference [96] is to our old friend Matthys, so there is some hope

MichaelG.

[S KParticipant @sk20060]

At the moment I trust John Haines info , even if the result entices me to stop this nonsense before I have even started..

Yes, the rule here is "always trust John Haines." 😉

[John HaineParticipant @johnhaine32865]

Well Philip Woodward was a respected mathematician as well as horologist and I think we can trust him!

If you take my expression:

An/A0 = exp(-n*pi/Q)

If An/A0 = 1/2 then exp(n*pi/Q) = 2

Or n*pi/Q = ln(2)

Q = n*pi/ln(2) which is Woodward's formula.

Edited By John Haine on 16/08/2023 17:08:47

Edited By John Haine on 16/08/2023 17:34:08

[S KParticipant @sk20060]

Posted by S K on 16/08/2023 15:43:18:

At the moment I trust John Haines info , even if the result entices me to stop this nonsense before I have even started..

Yes, the rule here is "always trust John Haines." 😉

Aaaand I misspelled John Haine. Sorry.

[John HaineParticipant @johnhaine32865]

By the way Joe, I love your angle sensor!

[Joseph Noci 1Participant @josephnoci1]

Wiki says:

The Q is related to how long it takes for the oscillations of an oscillator to die out. The Q of a pendulum can be measured by counting the number of oscillations it takes for the amplitude of the pendulum's swing to decay to 1/e = 36.8% of its initial swing, and multiplying by 'π.

John says ( and I saw it in two other papers…none of them written by John..)

Count cycles until amplitude is 63.2%, multiply by pi.

Those two add up to 100 for some useless reason, but each give very different results!

[duncan webster 1Participant @duncanwebster1]

I suspect the confusion over Q is caused by clockies insistence on calling the time for the pendulum to swing from one side to the other the period, whereas real engineers would call the period the time to get from extreme left back to extreme left, ie the time for one complete cycle. Wkipedia's definition is per radian, and one engineer's cycle is 2PI radians, which is where Alan (Philip Woodward) gets the 2PI. John Haine at 12:24 pm seems to be totally awry, as exp(-1) is 0.3678, not 0.6321

If I take my reasoning and using clockie definition of period then :

Q=swings to 61%*(2*PI)

Q=swings to 50%*4.5324

Q=swings to 36.79%*PI

Q=swings to 21%*2.013

which agrees with some of the figures JoNo is quoting

Edited By duncan webster on 16/08/2023 18:11:57

Edited By duncan webster on 16/08/2023 18:18:59

Edited By duncan webster on 16/08/2023 18:22:03

[SillyOldDufferModerator @sillyoldduffer]

Posted by Joseph Noci 1 on 16/08/2023 14:53:01:

Posted by S K on 16/08/2023 13:39:38:

I don't know where I got that formula from, but apparently it is wrong. And that would mean that my measured Q just got down-graded too. 😒

Everyone seems to be using a different formula or different method. This needs to be straightened out.

That's what I said 10 days ago…

Wait for amplitude to:

Drop by 21%, then times 2 =Q

Drop to 36%, then x 2 then x pi = Q

Drop til 50%, then x 4.53.

Drop till 63%, then times pi

The last 4 runs I did on my pendulum I captured swing counts at every 2% drop.

…

And the loose use of terms is also messy – Swings, periods, double swings, beats…..

Rawlings in 'The Science of Clocks and Watches', spends Chapter 4 on 'Dissipation of Energy by a Swing Pendulum'. He addresses the amount of energy needed to keep a pendulum swinging, but doesn't relate it to Q, which was new to clocks when he wrote circa WW2. Rawlings measures the decay of an actual pendulum and graphs it, Then from the graph he derives an equation that's a close match to the curve. The formula uses e (Naperian Log) and a constant of 16.6e10-5 per second; "chosen to make the curve pass as close to the squares as possible".

The 3rd Edition of Rawlings (1993) was updated and annotated by several up-to-date contributors, notably D A Bateman, who introduces Q. He mentions Drop until 50%, then x 4.53 as a way of estimating Q and explains that it and Joe's other values derive from the Naperian Log part of the equation. which assumes the pendulum exhibits 'damped harmonic motion', in much the same way as an early spark transmitter. DAB extends the maths in Appendix 2 'The Linear Theory of Vibration', which finishes with:

• The logarithmic decrement leads to an easy rule of thumb for measuring Q.
• Then: Q = πn/ln2 = 4.532n (approx) and
• With the low frequency of a second pendulum, this may entail waiting for an hour or more , but the calculation could hardly be simpler.

But I suggest there are several ways in which the formula could give wobbly results! For example, when a pendulum swings through a tiny arc, how accurately can the observer judge when 50% decay has occurred? And does he know that his pendulum is following 'damped harmonic motion'.

Unless the measurement is made very carefully in the same way by all parties, I think it unwise to get excited comparing the Q of other folks pendula. However, when the same method is applied consistently by the same operator, Q is useful as a way of checking whether a modification has improved or made a pendulum worse worse.

As a self-check, if the values of Q measured in the same run at 21%, 36%, 50% and 63% are wildly different, it's likely that the operator is doing it wrong, or that the pendulum is wonky, or both!

My method of deriving Q from a bandwidth formula has other problems. It assumes that the values of a large set of frequencies are normally distributed. This way of obtaining Q doesn't assume a logarithmic decay, which is just as well because my pendulum is being impulsed: definitely not 'damped harmonic motion'. And I'd expect the Q of a powered pendulum to be different from the Q of the same pendulum left to lose energy naturally. Nonetheless Bandwidth Q is meaningful to me because exactly the same method is applied every time I log new data, and I can trust it in my context. It's not necessarily useful to anyone else unless they use the same method.

Measuring is hard!!!

Dave

[John HaineParticipant @johnhaine32865]

Posted by duncan webster on 16/08/2023 18:11:07:

…John Haine at 12:24 pm seems to be totally awry, as exp(-1) is 0.3678, not 0.6321

…

Edited By duncan webster on 16/08/2023 18:11:57

Edited By duncan webster on 16/08/2023 18:18:59

Edited By duncan webster on 16/08/2023 18:22:03

Oops! Quite correct Duncan. Told you to trust Woodward.

[duncan webster 1Participant @duncanwebster1]

Posted by SillyOldDuffer on 16/08/2023 18:23:38:

Posted by Joseph Noci 1 on 16/08/2023 14:53:01:

Posted by S K on 16/08/2023 13:39:38:

I don't know where I got that formula from, but apparently it is wrong. And that would mean that my measured Q just got down-graded too. 😒

Everyone seems to be using a different formula or different method. This needs to be straightened out.

That's what I said 10 days ago…

Wait for amplitude to:

Drop by 21%, then times 2 =Q

Drop to 36%, then x 2 then x pi = Q

Drop til 50%, then x 4.53.

Drop till 63%, then times pi

The last 4 runs I did on my pendulum I captured swing counts at every 2% drop.

…

And the loose use of terms is also messy – Swings, periods, double swings, beats…..

 

………..

 

 

I think SOD is confusing drop BY and drop TO

Edited By duncan webster on 16/08/2023 18:46:19

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