A not-so-new Pendulum formula

As we have several members interested in Pendulum Clocks, I thought I would share this [which I only stumbled-across today] : **LINK**

http://leapsecond.com/hsn2006/pendulum-period-agm.pdf

MichaelG.

Fascinating.

That's brilliant. I hadn't heard of a geometric average before. Everyday's a school day!

I did a quick check ... and he’s right about the Means converging in just a few iterations

So this looks like it might be very useful indeed

MichaelG.

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< click for larger image >

I'm being picky but:

Even though the formula is exact as soon as you start calculating the mean of a rational and an irrational number it loses this exactness. It is only exact as long as you compute the square roots to an infinite number each time (unless the number is a perfect square to start with). As long as the irrational numbers are truncated it is still an approximation. It may be a good and useable one but perhaps this approximation in the calculation should have been pointed out in the written paper.

Martin C

It's worth noting that for practical clock pendulum amplitudes, say up to 10 degrees, the difference between the AGM formula and the usual "A^2/16" approximation to the circular deviation is very very small: 3.3 usec for a 1 second period and 10 degrees. I've done pendulum simulations for the order of amplitude used in "Clock B" and compared them to simple formulas based on the same approximation with virtually identical results.

Where the AGM really is useful is for methods where you need to quickly compute elliptic integrals. One example is designing very selective LC filters, where there are formulas for the component values that incorporate elliptic integrals. Once you needed to have extensive tables of these, but now they can be calculated more accurately in a few lines of code.

Very true, Martin .... but this does get us [with very little effort] several orders of magnitude beyond the classic ‘simplified formula’ which only works for small swings.

What excites me about it is that it’s accurate [or trivially inaccurate] for large angles.

MichaelG.

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P.S. __ The title of the original 2008 paper [see Further reading] is:

Approximations for the period of the simple pendulum based on the arithmetic-geometric mean

Edit: ___ The link appears to be dead, but this one works for me:

https://www.researchgate.net/publication/228353411_Approximation_for_the_period_of_the_simple_pendulum_based_on_the_arithmetic-geometric_mean

Edited By Michael Gilligan on 09/01/2021 10:55:24

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I am particularly interested in larger swings than that, John

Huygens, Harrison, et al ... [pendulum has ‘dominion’ and all that]

MichaelG