# Free harmonic oscillation

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

$$\tag{1 } x ( t) = \ A \cos ( \omega t + \phi ),$$

then it is said that $x ( t)$ performs a free harmonic oscillation. Here $A > 0$, $\omega > 0$ and $\phi$ are real constants, called, respectively, the amplitude, frequency and phase of the free harmonic oscillation. The period is $T = 2 \pi / \omega$. 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 $\omega t + \phi$ is called the instantaneous phase, and $\phi$ is called the initial phase. The quantity $\omega$ is also called the circular or cyclic frequency, and $f = \omega /2 \pi$ is called the frequency. A free harmonic oscillation (1) can be written as

$$x ( t) = \ a \cos \omega t + b \sin \omega t,$$

where $a, b$ and $A, \phi$ are connected by the relations

$$A = \ \sqrt {a ^ {2} + b ^ {2} } ,\ \ \cos \phi = \ { \frac{a}{\sqrt {a ^ {2} + b ^ {2} } } } ,\ \ \sin \phi = \ { \frac{b}{\sqrt {a ^ {2} + b ^ {2} } } } ,$$

or by

$$x ( t) = \ \mathop{\rm Re} ( Ae ^ {i ( \omega t + \phi ) } ).$$

Often the phase is taken to be $- \phi$ and not $\phi$.

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

$$\dot{x} dot + \omega ^ {2} x = 0.$$

For a mathematical pendulum of length $l$ and mass $m$, $\omega ^ {2} = g/l$; for a load of mass $m$ on a string with elasticity coefficient $k$, $\omega ^ {2} = k/m$; for an oscillating electrical circuit of capacity $C$ and inductance $L$, $\omega ^ {2} = 1/CL$. The equilibrium position $x = 0$, $\dot{x} = 0$ in the phase plane $( x, \dot{x} )$ for a free harmonic oscillator is the centre, and the phase trajectories are circles.

The sum $x _ {1} ( t) + x _ {2} ( t)$ of two free harmonic oscillations, where

$$x _ {j} ( t) = \ A _ {j} \cos \ ( \omega _ {j} t + \phi _ {j} ),\ \ j = 1, 2,$$

with commensurable frequencies $\omega _ {1}$ and $\omega _ {2}$ is a free harmonic oscillation. If $\omega _ {1}$ and $\omega _ {2}$ are incommensurable, then $x _ {1} ( t) + x _ {2} ( t)$ is an almost-periodic function, and

$$\sup _ {t \in \mathbf R } \ ( x _ {1} ( t) + x _ {2} ( t)) = \ A _ {1} + A _ {2} = \ - \inf _ {t \in \mathbf R } \ ( x _ {1} ( t) + x _ {2} ( t)).$$

The sum of $n$ free harmonic oscillations with rationally-independent frequencies $\omega _ {1} \dots \omega _ {n}$ is also almost-periodic. For the sum of two free harmonic oscillations, $\Omega = | \omega _ {1} - \omega _ {2} |$ is called the derangement. If $\Omega$ is small, $\Omega / \omega _ {1} \ll 1$, and if $\omega _ {1}$ and $\omega _ {2}$ have the same order of magnitude, then

$$x _ {1} ( t) + x _ {2} ( t) = \ A ( t) \cos ( \omega _ {1} t + \phi ( t)),$$

$$A ^ {2} ( t) = A _ {1} ^ {2} + A _ {2} ^ {2} + 2A _ {1} A _ {2} \cos ( \psi ( t) - \phi _ {1} ),$$

$$\psi ( t) = \Omega t + \phi _ {2} .$$

The "amplitude" $A ( t)$ is a slowly-varying function of period $2 \pi / \Omega$, and $A ^ {2} ( t)$ varies from $( A _ {1} - A _ {2} ) ^ {2}$ to $( A _ {1} + A _ {2} ) ^ {2}$. The oscillation $x _ {1} ( t) + x _ {2} ( t)$ is called a beat, and the "amplitude" $A ( t)$ alternatingly increases and decreases. This case is important in the analysis of receiving devices.

Suppose one has a system of $n$ equations

$$M \dot{x} dot + Kx = 0,\ \ x \in \mathbf R ^ {n} ,$$

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

$$\dot{y} dot _ {j} + \omega _ {j} ^ {2} y _ {j} = 0,\ \ j = 1 \dots n.$$

The coordinates $y _ {1} \dots y _ {n}$ are called normal. In normal coordinates $x ( t)$ 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)
How to Cite This Entry:
Free harmonic oscillation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_harmonic_oscillation&oldid=46984
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article