Van der Pol and the history of relaxation oscillations

Jean-Marc Ginoux, Christophe Letellier

Relaxation oscillations are commonly associated with the name of Balthazar van der Pol via his eponymous paper (Philosophical Magazine, 1926) in which he apparently introduced this terminology to describe the nonlinear oscillations produced by self-sustained oscillating systems such as a triode circuit.
Our aim is to investigate how relaxation oscillations were actually discovered. Browsing the literature from the late 19th century, we identified four self-oscillating systems in which relaxation oscillations have been observed:
i) the series dynamo machine conducted by Gerard-Lescuyer (1880),
ii) the musical arc discovered by Duddell (1901) and investigated by Blondel (1905),
iii) the triode invented by de Forest (1907)
and, iv) the multivibrator elaborated by Abraham and Bloch (1917).
The differential equation describing such a self-oscillating system was proposed by Poincare for the musical arc (1908), by Janet for the series dynamo machine (1919), and by Blondel for the triode (1919). Once Janet (1919) established that these three self-oscillating systems can be described by the same equation, van der Pol proposed (1926) a generic dimensionless equation which captures the relevant dynamical properties shared by these systems. Van der Pol’s contributions during the period of 1926-1930 were investigated to show how, with Le Corbeiller’s help, he popularized the “relaxation oscillations” using the previous experiments as examples and, turned them into a concept….
… Read more at http://arxiv.org/pdf/1408.4890v1.pdf

The laws of planetary motion …

… derived from those of a harmonic oscillator

KEPLERP. A. Horvathy
Kepler’s laws are deduced from those valid for a harmonic oscillator, following the approach of Bohlin, Levi-Civita and Arnold

1 Introduction
Kepler’s laws of planetary motion state that
1. K-I: A planet moves on an ellipse, one of whose foci being occupied by the sun;
2. K-II: The vector drawn from the sun to the planet’s position sweeps equal areas in equal
times
3. K-III: The squares of the periods are as the cubes of the major axes of the ellipses.
These laws can be deduced from the inverse-square force law and Newton’s equations of
motion…
… Read more at http://arxiv.org/pdf/1404.2265v1.pdf

Self-oscillation

Alejandro Jenkins

Illustration of the turbulent vortices generated by a flow of velocity v as it hits a circular obstacle of diameter d, on the left

Physicists are very familiar with forced and parametric resonance, but usually not with self-oscillation, a property of certain linear systems that gives rise to a great variety of vibrations, both useful and destructive. In a self-oscillator, the driving force is controlled by the oscillation itself so that it acts in phase with the velocity, causing a negative damping that feeds energy from the environment into the vibration: no external rate needs to be tuned to the resonant frequency. A paper from 1830 by G. B. Airy gives us the opening to introduce self-oscillation as a sort of “perpetual motion” responsible for the human voice. The famous collapse of the Tacoma Narrows bridge in 1940, often attributed by introductory physics texts to forced resonance, was actually a self-oscillation, as was the more recent swaying of the London Millenium Footbridge. Clocks are self-oscillators, as are bowed and wind musical instruments, and the heartbeat. We review the criterion that determines whether an arbitrary linear system can self-oscillate and describe the operation of two thermodynamic self-oscillators, the putt-putt toy boat and the Rijke tube, before concluding with a brief discussion of the relevance of the concept of self-oscillation to the semi-classical theory of lasers….
Read more: http://arxiv.org/PS_cache/arxiv/pdf/1109/1109.6640v1.pdf

Controlling Microscopic Friction through Mechanical Oscillations

A new technique could reduce friction with the mere flick of a switch, say physicists

The usual way to overcome friction is with a liberal coating of lubricant, whether it be some kind of oily liquid or a solid such as graphite.
But that’s a distinctly macroscopic solution. In recent years, however, physicists have become increasingly concerned with friction on the nanoscale. The problems arise in devices such as computer disc drives which can suffer stick-slip friction that is hard to overcome.
Today, Andrea Vanossi, at the University of Modena and Reggio Emilia, and a couple of pals investigate an interesting way of reducing friction on the nanoscale.
The idea, which has been around for a few years now, is to shake the surfaces involved. That makes sense on an intuitive level but exactly how this might reduce friction has never been fully investigated.
Vanossi and co examine the behaviour of a single tip, such as an atomic force microscope tip, in contact with a one-dimensional surface. In ordinary circumstances, the tip and the atoms on the surface arrange themselves in a way that minimises their energy. It is this energy barrier that causes stick-slip friction.
Overcoming friction is really a question of overcoming this barrier.
That’s where the oscillations turn out to be important. Shake the surface (or the tip) and this immediately raises the tip out of this minimum, allowing it to explore the energy landscape. This is equivalent to smooth sliding, or at least smoother sliding. Vanossi and co study the relationship between the friction and vibrations of various different frequencies and amplitudes.
So the vibration dramatically reduces friction. In fact, it essentially allows friction to be switched on and off.
But Vanossi and co have another interesting result. They say that once the oscillations have overcome stick-slip friction, they can help to maintain motion. In effect, the tip can ride the oscillations, like a surfer rides ocean waves.
That raises the prospect that the same mechanism that reduces friction could also help to move particles around surfaces, with appropriately designed waves.
That could turn out to be especially important for microelectromechanical devices. MEMS were once heralded as machines that would change the world but we are still waiting for this revolution largely because of the problem of ‘stiction’. Very often, these machines stick and remain stuck.
It’s just possible that a little careful shaking could help.
Ref: arxiv.org/abs/1106.2430: Controlling Microscopic Friction through Mechanical Oscillations
http://www.technologyreview.com/blog/arxiv/26885/