Hi Sam. It would be worth a try. Sometimes there is friction in the pendulum or gearwork mechanism and a bob with more mass would have more momentum that might overcome the frictional effects that are not considered, or insufficiently accounted for, in theoretical results. Good luck, I hope it helps. Let me know either way. — Ron

Forced Chaotic Pendulum Paper Description:
A sinusoidally-forced, large-amplitude pendulum was designed and built to demostrate the chaotic behavior that can arise in a simple nonlinear system. Following a brief review of the terminology associated with the study of chaos, the design of the forced pendulum, dubbed The Chaos Machine, is presented. Special attention is given to the electronic control system used to produce the sinusoidally varying torque that drives the pendulum. Standard frequency response techniques and time-domain simulations are used in the design of the control system. Finally, typical responses of the pendulum are presented that demonstrate phase-locked periodic and chaotic nonperiodic motion. The Chaos Machine promises to be a useful tool for teaching undergraduate students about nonlinear system dynamics.

Thanks, Fred! I read through your entire thesis and found it to be the clearest explanation of chaos theory that I have encountered! I’ve never had detailed knowledge of chaos theory, so it was refreshing to learn about it from such a well-written (and apparently letter-perfect!) source. The reason that a simple pendulum has a linear response at small angles is very apparent from your explanation of the difference between the approximate linear and exact non-linear differential equations governing it (the use of theta rather than the actual sin(theta) in the equation).

Thanks, Benji. I haven’t posted lately but I’ve been very busy working on these types of things so there are essays in the pipeline. — Ron

Hi Donald. I’m sorry, but I don’t have any suggestions for you. If anyone else here who is familiar with this clock has a suggestion, please leave a comment here. Thanks. — Ron

http://www.hsn161.com/hsn_article.php

http://www.telegraph.co.uk/news/obituaries/1337670/Professor-E-T-Teddy-Hall.html

It used an elaborate computerized control system that monitored the motions of the pendulum with an LED and photocell so that nothing needed to physically contact the pendulum and an electromagnet gave it precisely enough energy to keep its swing at a constant amplitude. Temperature was controlled to a fraction of a degree Celsius. To reduce the effect of vibrations, he built an elaborate support structure with about 12 tonnes of concrete. He managed accuracy about an order of magnitude better than the Shortt free pendulum clock of the 1920s
http://en.wikipedia.org/wiki/Escapement#Free_pendulum_clock

Wow, his clock simply amazing to read about. At the moment I’m reading “My Own Right Time: An Exploration of Clockwork Design” by Philip Woodward, from a recommendation of an horologist who contacted me. What a great book–it’s his story of how he started out knowing very little about pendulum clocks and his failures and successes in trying to design the most accurate one he could. It’s extremely readable with a light air, while delivering a great deal of technical information. (It’s also extremely expensive new at Amazon, so mine’s through interlibrary loan!) — Ron

The name of the Russian was Feodsii Michailovich Fedchenko. The path followed by the centre of mass of his pendulum has not been determined but it probably was not cycloidal.
All pendulum clocks can sense the lunar solar efffect ie tides. It’s just that the variation is swamped by other sources of instability.

You will see some papers on these subjects on my website.

Incidentally I reckon the term used to be deduced reckoning. Dead reckoning is a coruption.

Alan

Thanks for the correction on the barometric effects, Alan. I’ve updated the essay to incorporate your changes as well as some I received from others via email. I’ve also read that “dead reckoning” comes from “deduced reckoning” and here I’m corrupting it even more with wordplay by implying dead forms of math reckoning. I’ve also added your website to the list of references at the end of the essay—your papers on that site are fascinating, just the kind of thing I’m aiming for in this blog! — Ron

I’ve just submitted that other paper on the large-angle pendulum period to Am. J. Phys. This complete version will soon be posted on the Cornell “arxiv” and then you can post a copy on your WebSite. You will surely be glad to know that I included you at the Acknowledgments, at the end of the paper.
After some days of rest I’ll try to solve some (several)century-old problems in Mathematics, namely the determination of a closed-form for zeta(3) = sum(1/(n^3), n=1..infinity) and also the Catalan constant [=sum((-1)^(n-1)/((2n-1)^2), n=1..infinity)]. I think this will demand at least one year. In the meantime I intend to treat some other problems in quantum physics.
Thanks,

Fabio M. S. Lima

Thanks for the pointer! I’ve read through your paper (which can be downloaded as http://arxiv.org/vc/physics/papers/0510/0510206v1.pdf) and you’re absolutely correct. For the information of others here, the approximation I provided (from the truncation of a series by Bernoulli) is calculated in the paper as having an error of 0.1% and 0.5% for amplitudes of 41° and 60°, respectively. Dr. Lima derived a logarithmic approximation by linear interpolation of the denominator in the elliptic integral of the exact solution, yielding T = -T₀ ln(a)/(1-a), where T₀ is the small-angle formula and a = cos(θ/2). This formula exhibits an error of 0.1% and 0.2% for amplitudes of 74° and 86°, respectively. This is a significant improvement for such a simple formula, and as the paper points out, this is increasingly important as today’s electronic timers and detectors are available to students in physics labs. Thanks again for taking the time to comment on this. — Ron

I hadn’t heard of this. Alan Emmerson also refers to it in a later comment below. — Ron

Thanks

Hi Gregg. All other things being equal, the period of a pendulum increases as the bob moves further from the pivot point, so the clock slows down. The approximate equation for the period T of a pendulum in terms of the acceleration g due to earth’s gravity, the constant pi, and the length L (measured from the pivot point to the center of mass) is T = 2π(L/g)^1/2, so the period increases as the square root of the length. In other words, if we assume all the mass is located in the bob, doubling its distance will increase its period by a factor of 2^1/2=1.414. The exact equation for the period of a pendulum is not expressible as a finite equation like this, but it turns out that the period calculated from it also varies directly with the square root of L. Perhaps your inconsistent results are due to changes happening in the amplitude of the pendulum swing when you move the bob—larger swings of a pendulum increase its period, and here the exact equation can be approximated as T = T₀(1 + θ²/16), where T₀ is the small-angle period calculated from the earlier formula and θ is the amplitude (angle) of the swing. So if the swing increases as a side effect of moving the bob closer to the pivot point, then they will have opposing effects on the period that can make understanding what is going on much more difficult. — Ron D.

Thanks, my first pingback—Ron D.