Oscillation

Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states.

Familiar examples of oscillation include a swinging pendulum and alternating current.

Oscillations occur not only in mechanical systems but also in dynamic systems in virtually every area of science: for example the beating of the human heart (for circulation), business cycles in economics, predator–prey population cycles in ecology, geothermal geysers in geology, vibration of strings in guitar and other string instruments, periodic firing of nerve cells in the brain, and the periodic swelling of Cepheid variable stars in astronomy.

If a constant force such as gravity is added to the system, the point of equilibrium is shifted.

The systems where the restoring force on a body is directly proportional to its displacement, such as the dynamics of the 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 system, oscillations occur because, at the static equilibrium displacement, the mass has kinetic energy which is converted into potential energy stored in the spring at the extremes of its path.

In the case of the spring-mass system, Hooke's law states that the restoring force of a spring is:

The solution to this differential equation produces a sinusoidal position function:

where ω is the frequency of the oscillation, A is the amplitude, and δ is the phase shift of the function.

This means there are dissipative processes such as friction or electrical resistance which continually convert some of the energy stored in the oscillator into heat in the environment.

Thus, oscillations tend to decay with time unless there is some net source of energy into the system.

Damped oscillators are created when a resistive force is introduced, which is dependent on the first derivative of the position, or in this case velocity.

The differential equation created by Newton's second law adds in this resistive force with an arbitrary constant b.

The exponential term outside of the parenthesis is the decay function and β is the damping coefficient.

In addition, an oscillating system may be subject to some external force, as when an AC circuit is connected to an outside power source.

The simplest example of this is a spring-mass system with a sinusoidal driving force.

This transfer 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 still greater displacement.

At sufficiently large displacements, the stiffness of the wing dominates to provide the restoring force that enables an oscillation.

The harmonic oscillator and the systems it models have a single degree of freedom.

For example, two pendulum clocks (of identical frequency) mounted on a common wall will tend to synchronise.

[2] The apparent motions of the compound oscillations typically appears very complicated but a more economic, computationally simpler and conceptually deeper description is given by resolving the motion into normal modes.

Depending on the starting point of the masses, this system has 2 possible frequencies (or a combination of the two).

In this case the regions of synchronization, known as Arnold Tongues, can lead to highly complex phenomena as for instance chaotic dynamics.

The force that creates these oscillations is derived from the effective potential constant above:

This differential equation can be re-written in the form of a simple harmonic oscillator:

By thinking of the potential in this way, one will see that at any local minimum there is a "well" in which the ball would roll back and forth (oscillate) between

As the number of degrees of freedom becomes arbitrarily large, a system approaches continuity; examples include a string or the surface of a body of water.

Such systems have (in the classical limit) an infinite number of normal modes and their oscillations occur in the form of waves that can characteristically propagate.

The mathematics of oscillation deals with the quantification of the amount that a sequence or function tends to move between extremes.