Free harmonic oscillation

A sinusoidal oscillation. If a mechanical or physical quantity , where denotes time, varies in accordance with the law

(1)

then it is said that performs a free harmonic oscillation. Here , and are real constants, called, respectively, the amplitude, frequency and phase of the free harmonic oscillation. The period is . The following terminology is often used in physics and engineering. A free harmonic oscillation is called a harmonic oscillation, or a simple harmonic oscillation; functions of the form (1) are called harmonics, the variable is called the instantaneous phase, and is called the initial phase. The quantity is also called the circular or cyclic frequency, and is called the frequency. A free harmonic oscillation (1) can be written as

where and are connected by the relations

or by

Often the phase is taken to be and not .

Small oscillations of mechanical or physical systems with one degree of freedom near a stable non-degenerate equilibrium position are free harmonic oscillations, to a large degree of exactness. For example, small oscillations of a pendulum, oscillations of a load suspended by a string, oscillations of a tuning fork, the variation of the direction and strength of the current in an oscillating electrical circuit, the rolling of a ship, etc. A system performing free harmonic oscillation is called a linear harmonic oscillator, and its oscillation is described by the equation

For a mathematical pendulum of length and mass , ; for a load of mass on a string with elasticity coefficient , ; for an oscillating electrical circuit of capacity and inductance , . The equilibrium position , in the phase plane for a free harmonic oscillator is the centre, and the phase trajectories are circles.

The sum of two free harmonic oscillations, where

with commensurable frequencies and is a free harmonic oscillation. If and are incommensurable, then is an almost-periodic function, and

The sum of free harmonic oscillations with rationally-independent frequencies is also almost-periodic. For the sum of two free harmonic oscillations, is called the derangement. If is small, , and if and have the same order of magnitude, then

The "amplitude" is a slowly-varying function of period , and varies from to . The oscillation is called a beat, and the "amplitude" alternatingly increases and decreases. This case is important in the analysis of receiving devices.

Suppose one has a system of equations

where and are real symmetric positive-definite matrices with constant elements. By using an invertible transformation this system is transformed into the decomposed system

The coordinates are called normal. In normal coordinates is the vector sum of free harmonic oscillations along the coordinate axes.

References

[1]	A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Dover, reprint (1987) (Translated from Russian)
[2]	G.S. Gorelik, "Oscillations and waves" , Moscow-Leningrad (1950) (In Russian)
[3]	L.D. Landau, E.M. Lifshits, "Mechanics" , Pergamon (1965) (Translated from Russian)