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
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where and
are connected by the relations
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or by
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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
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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
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with commensurable frequencies and
is a free harmonic oscillation. If
and
are incommensurable, then
is an almost-periodic function, and
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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
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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
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where and
are real symmetric positive-definite matrices with constant elements. By using an invertible transformation
this system is transformed into the decomposed system
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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) |
Free harmonic oscillation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_harmonic_oscillation&oldid=17959