Oscillation

Oscillation is the periodic variation, typically in time, of some measure as seen, for example, in a swinging pendulum. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes is used to be synonymous with oscillation. Oscillations occur not only in physical systems but also in biological systems and in human society. Oscillations are the origin of the sensation of musical tone.

Table of contents

1 Simple systems
2 Damped, driven and self-excited oscillations

2.1 Driven oscillations
2.2 Self-exciting systems

3 Coupled oscillations
4 Continuous systems - waves
5 Examples

5.3 Mechanical
5.4 Electrical
5.5 Electro-mechanical
5.6 Biological
5.7 Human
5.8 Economic and social
5.9 Climate and geophysics

Simple systems

The simplest oscillating system is a mass, subject to the force of gravity, attached to a linear spring. The system is in an equilibrium state when the weight of the mass is balanced by the tension of the spring. If the system is displaced from the equilibrium, there is a net restoring force on the mass, tending to bring it back to equilibrium. However, in moving the mass back to the equilibrium position, it has acquired inertia which keeps it moving beyond that position, establishing a new restoring force, now in the opposite sense. The specific dynamics of this spring-mass system are described mathematically by the simple harmonic oscillator and the regular periodic motion is known as simple harmonic motion. In the spring-mass sytem, oscillations occur because, when at the static equilibrium displacement, the mass has kinetic energy which is converted into energy stored in the spring at the extremes of its path.

The spring-mass system illustrates some important and universal principles of oscillation:

Existence of an equilibrium;
Presence of some restoring force (or restoring principle in non-mechanical systems);
Some form of "inertia" that maintains motion; and
Exchange in "energy" between that associated with "inertia" and that of the restoring force.

The harmonic oscillator offers a model of many more complicated types of oscillation and can be extended by the use of Fourier analysis.

Damped, driven and self-excited oscillations

In real-world systems, the second law of thermodynamics dictates that there is some continual and inevitable conversion of energy into the thermal energy of the environment. Thus, damped oscillations tend to decay with time unless there is some net source of energy in the system. The simplest description of this decay process can be illustrated by the harmonic oscillator.

Driven oscillations

One possible external source of energy is to drive the oscillations, as in exciting the mass-spring system by periodically moving the spring's anchor point. The driving force can be structured, perhaps itself a simple harmonic motion, or random. In general, the response of the system is not independent of the period of a simple-harmonic driving force. The amplitude of the system's response tends to be highly-peaked at particular periods, the phenomenon of resonance.

Self-exciting systems

Some systems are able to extract energy from their environment. This typically occurs where systems are embedded in some fluid flow. For example, the phenomenon of flutter in aerodynamics occurs when an, arbitrarily small, displacement of an aircraft wing, from its equilibrium, results in an increase in the angle of attack of the wing on the air flow and a consequential increase in lift coefficient leading to a greater displacement before, at sufficiently large displacements, the stiffness of the wing dominates to provide the restoring force that enables an oscillation. The phenomenon of hunting is a self-exciting oscillation in non-linear electrical circuits.

Coupled oscillations

The harmonic oscillator, and the more complicated systems for which it stands as a simple model, has a single degree of freedom. More complicated systems have more degrees of freedom, for example two masses and two springs. In such cases, energy is converted between the respsective inertias of each degree of freedom and the several restoring forces in the system. This leads to a coupling of the oscillations of the individual degrees of freedom. For example, two pendulum clocks mounted on a common wall will tend to synchronise. The apparent motions of the individual oscillations typically appears very complicated but a more economic, computationally simpler and conceptually deeper description is given by resolving the motion into normal modes.

Continuous systems - waves

As the number of degrees of freedom becomes indefinitely large, a system approaches continuity , for example, a string or the surface of a body of water. Such systems have an infinite number of normal modes and their oscillations occur in the form of waves that have the characteristic that they can propagate.