![Self excited oscillation](https://upload.wikimedia.org/wikipedia/commons/8/8e/Self_excited_oscillation.svg)
Self-oscillation is the generation and maintenance of a periodic motion by a source of power that lacks any corresponding periodicity. The oscillator itself controls the phase with which the external power acts on it. Self-oscillators are therefore distinct from
forced and
parametric resonators, in which the power that sustains the motion must be modulated externally.
In
linear systems
In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.
Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.
As a mathematical abstraction ...
, self-oscillation appears as an instability associated with a negative
damping
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
term, which causes small perturbations to grow exponentially in amplitude. This negative damping is due to a
positive feedback
Positive feedback (exacerbating feedback, self-reinforcing feedback) is a process that occurs in a feedback loop which exacerbates the effects of a small disturbance. That is, the effects of a perturbation on a system include an increase in the ...
between the oscillation and the modulation of the external source of power. The amplitude and waveform of steady self-oscillations are determined by the
nonlinear characteristics of the system.
Self-oscillations are important in physics, engineering, biology, and economics.
History of the subject
The study of self-oscillators dates back to
Robert Willis,
George Biddell Airy
Sir George Biddell Airy (; 27 July 18012 January 1892) was an English mathematician and astronomer, and the seventh Astronomer Royal from 1835 to 1881. His many achievements include work on planetary orbits, measuring the mean density of the E ...
,
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
, and
Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Amo ...
in the 19th century. The term itself (also translated as "auto-oscillation") was coined by the Soviet physicist
Aleksandr Andronov
Aleksandr Aleksandrovich Andronov (russian: Алекса́ндр Алекса́ндрович Андро́нов; , Moscow – October 31, 1952, Gorky) was a Soviet physicist and member of the Soviet Academy of Sciences (1946). He worked exten ...
, who studied them in the context of the mathematical theory of the
structural stability
In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact ''C''1-small perturbations).
Examples of such q ...
of
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s. Other important work on the subject, both theoretical and experimental, was due to
André Blondel
André-Eugène Blondel (28 August 1863 – 15 November 1938) was a French engineer and physicist. He is the inventor of the electromechanical oscillograph and a system of photometric units of measurement.
Life
Blondel was born in Chaumont, Ha ...
,
Balthasar van der Pol
Balthasar van der Pol (27 January 1889 – 6 October 1959) was a Dutch physicist.
Life and work
Van der Pol began his studies of physics in Utrecht in 1911. J. A. Fleming offered van der Pol the use of the Pender Electrical Laboratory at ...
,
Alfred-Marie Liénard
Alfred-Marie Liénard (2 April 1869 in Amiens – 29 April 1958 in Paris), was a French physicist and engineer. He is most well known for his derivation of the Liénard–Wiechert potentials.
From 1887 to 1889 Liénard was a student at the Éco ...
, and
Philippe Le Corbeiller
Philippe Emmanuel Le Corbeiller (January 11, 1891 – July 24, 1980) was a French-American electrical engineer, mathematician, physicist, and educator. After a career in France as an expert on the electronics of telecommunications, he became ...
in the 20th century.
The same phenomenon is sometimes labelled as "maintained", "sustained", "self-exciting", "self-induced", "spontaneous", or "autonomous" oscillation. Unwanted self-oscillations are known in the mechanical engineering literature as
hunting
Hunting is the human activity, human practice of seeking, pursuing, capturing, or killing wildlife or feral animals. The most common reasons for humans to hunt are to harvest food (i.e. meat) and useful animal products (fur/hide (skin), hide, ...
, and in electronics as
parasitic oscillation Parasitic oscillation is an undesirable electronic oscillation (cyclic variation in output voltage or current) in an electronic or digital device. It is often caused by feedback in an amplifying device. The problem occurs notably in RF, audio, ...
s.
Important early studied examples of self-oscillation include the
centrifugal governor
A centrifugal governor is a specific type of governor with a feedback system that controls the speed of an engine by regulating the flow of fuel or working fluid, so as to maintain a near-constant speed. It uses the principle of proportional cont ...
and railroad wheels.
Mathematical basis
Self-oscillation is manifested as a linear instability of a dynamical system's static
equilibrium. Two mathematical tests that can be used to diagnose such an instability are the
Routh–Hurwitz and
Nyquist criteria. The amplitude of the oscillation of an unstable system grows exponentially with time (i.e., small oscillations are negatively damped), until nonlinearities become important and limit the amplitude. This can produce a steady and sustained oscillation. In some cases, self-oscillation can be seen as resulting from a time lag in a
closed loop system, which makes the change in variable ''x
t'' dependent on the variable ''x
t-1'' evaluated at an earlier time.
Examples in engineering
Railway and automotive wheels
Hunting oscillation
Hunting oscillation is a self-oscillation, usually unwanted, about an equilibrium. The expression came into use in the 19th century and describes how a system "hunts" for equilibrium. The expression is used to describe phenomena in such diverse ...
in
railway
Rail transport (also known as train transport) is a means of transport that transfers passengers and goods on wheeled vehicles running on rails, which are incorporated in tracks. In contrast to road transport, where the vehicles run on a pre ...
wheel
A wheel is a circular component that is intended to rotate on an axle Bearing (mechanical), bearing. The wheel is one of the key components of the wheel and axle which is one of the Simple machine, six simple machines. Wheels, in conjunction wi ...
s and
shimmy
A shimmy is a dance move in which the body is held still, except for the shoulders, which are quickly alternated back and forth. When the right shoulder goes back, the left one comes forward.
History
In 1917, a dance-song titled "Shim-Me-Sha ...
in automotive
tire
A tire (American English) or tyre (British English) is a ring-shaped component that surrounds a Rim (wheel), wheel's rim to transfer a vehicle's load from the axle through the wheel to the ground and to provide Traction (engineering), t ...
s can cause an uncomfortable wobbling effect, which in extreme cases can derail trains and cause cars to lose grip.
Central heating thermostats
Early central heating
thermostat
A thermostat is a regulating device component which senses the temperature of a physical system and performs actions so that the system's temperature is maintained near a desired setpoint.
Thermostats are used in any device or system tha ...
s were guilty of self-exciting oscillation because they responded too quickly. The problem was overcome by
hysteresis
Hysteresis is the dependence of the state of a system on its history. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field changed in the past. Plots of a single component of ...
, i.e., making them switch state only when the temperature varied from the target by a specified minimum amount.
Automatic transmissions
Self-exciting oscillation occurred in early
automatic transmission
An automatic transmission (sometimes abbreviated to auto or AT) is a multi-speed transmission used in internal combustion engine-based motor vehicles that does not require any input from the driver to change forward gears under normal driving c ...
designs when the vehicle was traveling at a speed which was between the ideal speeds of 2 gears. In these situations the transmission system would switch almost continuously between the 2 gears, which was both annoying and hard on the transmission. Such behavior is now inhibited by introducing
hysteresis
Hysteresis is the dependence of the state of a system on its history. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field changed in the past. Plots of a single component of ...
into the system.
Steering of vehicles when course corrections are delayed
There are many examples of self-exciting oscillation caused by delayed course corrections, ranging from light aircraft in a strong wind to erratic steering of road vehicles by a driver who is inexperienced or drunk.
SEIG (self-excited induction generator)
If an
induction motor
An induction motor or asynchronous motor is an AC electric motor in which the electric current in the rotor needed to produce torque is obtained by electromagnetic induction from the magnetic field of the stator winding. An induction mot ...
is connected to a capacitor and the shaft turns above synchronous speed, it operates as a self-excited induction generator.
Self-exciting transmitters
Many early radio systems tuned their transmitter circuit, so the system automatically created radio waves of the desired frequency. This design has given way to designs that use a separate oscillator to provide a signal that is then amplified to the desired power.
Examples in other fields
Population cycles in biology
For example, a reduction in population of an herbivore species because of
predation
Predation is a biological interaction where one organism, the predator, kills and eats another organism, its prey. It is one of a family of common feeding behaviours that includes parasitism and micropredation (which usually do not kill the ...
, this makes the populations of predators of that species decline, the reduced level of predation allows the herbivore population to increase, this allows the predator population to increase, etc. Closed loops of time-lagged differential equations are a sufficient explanation for such cycles - in this case the delays are caused mainly by the breeding cycles of the species involved.
See also
*
Hopf bifurcation
In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses ...
*
Limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity ...
*
Van der Pol oscillator
In dynamical system, dynamics, the Van der Pol oscillator is a Conservative force, non-conservative oscillator with nonlinearity, non-linear Damping ratio, damping. It evolves in time according to the second-order differential equation:
:-\mu(1-x ...
*
Hidden oscillation
In the bifurcation theory, a bounded oscillation that is born without loss of stability of stationary set is called a hidden oscillation. In nonlinear control theory, the birth of a hidden oscillation in a time-invariant control system with bounde ...
References
{{Reflist.
Oscillators
Amplifiers
Systems theory
Dynamical systems
Nonlinear systems
Mechanical vibrations
Physical phenomena
Ordinary differential equations
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