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Dynamics and performance of clock pendulums: http://arxiv.org/abs/1501.03673

We analyze the dynamics of a driven, damped pendulum as used in mechanical clocks. We derive equations for the amplitude and phase of the oscillation, on time scales longer than the pendulum period. The equations are first order ODEs and permit fast simulations of the joint effects of circular and escapement errors, friction, and other disturbances for long times. The equations contain two averages of the driving torque over a period, so that the results are not very sensitive to the fine structure of the driving. We adopt a constant-torque escapement and study the stationary pendulum rate as a function of driving torque and friction. We also study the reaction of the pendulum to a sudden change in the driving torque, and to stationary noisy driving. The equations for the amplitude and phase are shown to describe the pendulum dynamics quite well on time scales of one period and longer. Our emphasis is on a clear exposition of the physics.

Rotating saddle trap as Foucault's pendulum: a hidden 'Coriolis' force in an inertial frame: http://arxiv.org/abs/1501.03658

According to Earnshaw's theorem an electrostatic potential cannot have stable equilibria, i.e. the minima, since such potentials are harmonic functions. However, the 1989 Nobel Prize in physics was awarded to W. Paul for his invention of the trap for suspending charged particles in an oscillating electric field. Paul's idea was to stabilize the saddle by "vibrating" the electrostatic field, by analogy with the so-called Stephenson-Kapitsa pendulum in which the upside-down equilibrium is stabilized by vibration of the pivot. Instead of vibration, the saddle can also be stabilized by rotation of the potential (in two dimensions); this has been known for nearly a century, since 1918. Particles confined in rotating saddle traps exhibit precession in the laboratory frame, which up to now has been explained by analyzing explicit solutions. Here we show that this precession is actually due to a hidden Coriolis-like force, which we uncover by a normal form transformation. Unlike the conventional Coriolis force, this one is not an inertial force, as it acts in the non-rotating frame. To our knowledge this is the first example of a Coriolis-like force arising in an inertial frame.


( 3 comments — Leave a comment )
Jan. 16th, 2015 08:27 pm (UTC)
Что-то мне некстати вспомнилось, что слово pendulum всегда вызывало у меня ассоциации с чем-то родным советским - это такая хуйня, которой надо отвалить могучий пендюлюм, чтобы она раскачалась.

О, заодно идея для фильма "Гарри Поттер и Гопники со Ждани": там должно быть заклинание "пендюлюм кирзачум!"
Jan. 16th, 2015 08:32 pm (UTC)
Да, пендулум - словечко смачное. Кстати, у Хэрриота в оригинале нет-нет, да и всплывает словечко penduling про болтающиеся ... какого-нибудь быка)))
(Deleted comment)
Jan. 17th, 2015 11:29 am (UTC)
На здоровье!
( 3 comments — Leave a comment )

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