Self-oscillation paper by Alejandro Jenkins

The existence of cycles may be considered as resulting from a small number of basic mechanisms. In this wide ranging paper on arXiv titled Self-oscillation, Alejandro Jenkins looks at one of these fundamental mechanisms:

Physicists are very familiar with forced and parametric resonance, but usually not with self-oscillation, a property of certain dynamical 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 into the vibration: no external rate needs to be adjusted to the resonant frequency. A little-known paper from 1830 by G. B. Airy provides us with 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 swaying of the London Millennium Footbridge in 2000. Clocks are self-oscillators, as are bowed and wind musical instruments. The heart is a “relaxation oscillator,” i.e., a non-sinusoidal self-oscillator whose period is determined by sudden, nonlinear switching at thresholds. We review the general criterion that determines whether a linear system can self-oscillate. We then describe the limiting cycles of the simplest nonlinear self-oscillators. We characterize the operation of motors as self-oscillation and prove a general theorem about their limit efficiency, of which Carnot’s theorem for heat engines appears as a special case. We also briefly discuss how self-oscillation applies to servomechanisms, Cepheid variable stars, lasers, and the macroeconomic business cycle, among other applications.

The paper is rather mathematical, but will benefit every student of cycles through its bringing together such a wide ranging set of systems under a single understanding. The summary is readable by all and begins with this statement:

Like the mythical perpetual motion machine, self-oscillation succeeds in driving itself, but does so in a way compatible with the known laws of physics.64 Instances of self-oscillation, both useful and destructive, abound in mechanical engineering, music, biology, electronics, and medicine. Indeed, all technology ultimately depends on self-oscillation, since only a self-oscillator can turn a steady source of power into a regular cyclical motion.

Self-oscillators, as distinct from forced and parametric resonators, can be readily identified by the fact that they sustain large, regular oscillations without an external rate having to be tuned to that frequency: the motion itself sets the phase of the driving force. We reviewed, for instance, how a violin string works as a self-oscillator, since increasing the velocity of the bow simply causes the same note to play more loudly, while the æolian harp, on the other hand, is a forced resonator, which rings loudly only when the wind speed happens to give a Strouhal frequency of vortex shedding close to the fundamental tone of the string (or to one of its harmonics).

In his early mathematical modeling of the vocal chords, Airy obtained self-oscillation from a delayed component of the harmonic restoring force. More generally, self-oscillation can be understood as the result of a component of the driving force that is modulated in phase with the velocity of the displacement. This gives the device a negative damping, causing the amplitude to grow exponentially with time, until nonlinear effects become significant.

It also has an extensive bibliography of more than 300 papers.


About Ray Tomes

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