Pendulum 'Q' value and measurement methods

If I estimate your "half power bandwidth" to be 2 times your standard deviation (that's the number I have, anyway), then after rounding the numbers for simplicity, I get Q=1/(2*0.001)=500. How do you get over 20,000? What numbers do you have?

Also, you expressed concern about how your cumulative time error had wandered. This is due in large part to the accumulation of many small random errors in a classic random walk. If those small errors are 250 times lower (i.e., S.D. = 4us vs. 1ms), then the cumulative deviations should, in an typical run, also be 250 times lower. So yes, you do want as low a standard deviation as you can get.

Edited By S K on 18/08/2023 01:46:06

This might be useful: **LINK**

http://www.vibrationdata.com/tutorials2/half_power_bandwidth.pdf

It’s a long while since I worked in vibration testing, but it looks right.

caveat lector

MichaelG.

I thought it would be interesting to directly compare the normal distribution curve with a resonance curve according to the equation Dave quoted above.

In this picture the normal (blue) curve is plotted using Excel's normal distribution function normalised so the peak is unity. I've normalised the mean to unity and the standard deviation to the value that Dave measured divided by his period, to give 0.001194. I've plotted the vertical axis in dB to make it easy to compare the 3dB points. I wasn't quite sure how to render the vertical probability axis in dB since it doesn't really have a meaning but I chose 20*log10 to match what I did with the resonance formula. The resonance curve (orange) used the same centre frequency of 1Hz and I twiddled the Q to to get the -3dB points to match the normal curve, which needed a Q of ~500.

What's obvious is that though the curves are similar around the peak they diverge hugely as one moves away, by 10s of dB.

It seems to me that the period error must depend on the Q, obviously, but also on the "noisiness" of the whole oscillator, and the amplitude of the oscillation which the noise is perturbing. Those factors are captured in the equation I posted. Clock B is a nice example - the pendulum has a rather low Q compared to most regulators, but it swings with about 4 - 6x the amplitude, the drive torque is very accurately controlled with a remontoire, the escapement minimises frictional variation of force, the whole thing is extremely massive and (for the critical tests) was mounted on a masonry column embedded in boulder clay; and finally the pendulum is temperature and barometrically compensated.

I think Dave's distribution looks very gaussian (normal) and is actually a very useful measure though not of Q! It tells us I think that the systematic variations are pretty small and most of the fluctuations are random, though they could be due either to the oscillator or the measurement system.

No point in denying it - I'm worried, and have been since John emailed a couple of his papers to me last week . Not quite convinced myself yet I'm wrong about Q but I trust John's judgement in this area. So I spent this afternoon bashing my brains in hope of understanding it!

Doesn't prove anything but I produced these similar graphs, also comparing normal and resonant curves:

First up is my pendulum's frequency distribution.

Second is my pendulum's frequency distribution fitted to a normal distribution. The match is good suggesting my pendulum is producing normally distributed periods.

Third is a resonant curve generated from this formula found under 'Universal Resonance Curve' in Wikipedia's Resonance article:

ω is the natural resonant frequency
Ω is the drive frequency
Γ is the decay factor

Damping the third curve with a decay factor of 0.1 produces a curve similar to the actual pendulum distribution

Fourth graph is the resonance curve generated with a lighter decay factor. As expected it produces a much sharper curve.

No wonder I'm confused!

Dave

Stepping gingerly as I'm not at all sure what I'm talking about, Dave's third and 4th curves relate to a resonant system being driven at a range of frequencies above and below the resonance. A pendulum is driven at its resonant frequency, or at say 1/15th for a Synchronome, or not at all when doing a run down test. I can't help feeling that these are 2 different scenarios. My pendulum keeps noticeable swinging for quite a long time if I turn the power off, can't exactly remember, but tens of minutes, does a inductor/capacitor manage this?

If I had a pendulum which was perfect in every way apart from a bit of damping and I let it run down, the frequency would very gradually increase as the amplitude decayed, but it wouldn't have a normal distribution.

An LC circuit of the same Q and resonant frequency as a pendulum would take the same number of cycles to decay and hence the same time.

The decay formula is :

However LC circuits usually oscillate at much higher frequencies than pendula, so the number of cycles needed for an LC resonator to decay to the same level occur much faster, An oscillator of Q=10000 at 10MHz decays 10 million times faster than one of Q=10000 oscillating at 1 Hz.

Dave

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With the greatest respect and empathy, Dave … I think even a damping factor of 0.01 [your fourth graph] would be appallingly high for a pendulum.

MichaelG.

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Edit: __ in the case for the prosecution, I present this [found after I posted] well-documented, and easily replicated, experiment:  https://arxiv.org/pdf/2002.03796.pdf

Edited By Michael Gilligan on 18/08/2023 20:32:22

The period would decrease very slightly, but the cycle-to-cycle random variation would be very small unless there was a lot of noise affecting it (for example support vibration).

… For appallingly, please read rather

MichaelG.

Not sure about the use of the general resonance formula as applied to an intermittently driven oscillator. ?

regards Martin

Oh dear, a major breakdown of communications! No wonder - it's getting complicated,

My Graphs 3 and 4 are examples illustrating how the shape of a universal resonance curve depends on the damping factor. They're a response to John's post in which he compares a normal distribution curve to a resonant curve and suggests the curves move away from each other. I'm pointing out by example that the shape of a resonance curve can be made to look like a normal distribution by tweaking the decay factor. I said I don't believe it proves anything!

This part of the discussion relates to a serious criticism John is making of the way I calculate Q-factor using what I believe to be the valid bandwidth method. Serious because I'm an amateur poking around in the dark, whilst this is John's area of expertise and he can do the maths! At the moment I'm in the unhappy position of not understanding my own work or understanding why John thinks I've got it wrong.

My efforts this afternoon have shown my pendulum's frequency distribution is close to a normal distribution and that I can synthesise a resonance curve to take much the same shape. I'm no nearer proving to my own satisfaction that the way I calculate Q is right or wrong. Sadly, when an expert tells an amateur he's messed up, the expert is usually right...

Dave

An interesting development this afternoon! I found a code error. The standard deviation of my data isn't about 1mS, it's 0.037 milliseconds, which by SK's method above gives my pendulum Q=13500.

Sorry about that, my fault.

Assuming the data and method I'm using is correct, which has been strongly challenged by John, the 3db bandwidth of my pendulum is about 0.044 milliseconds, period 0.937357s.

Dave

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mea culpa …

but I would still recommend the brief paper that I linked.

MichaelG.