Harmonic vibration

sinusoidal vibration

A periodic variation with time of a physical magnitude, which may be written in analytical form as

$$ x = x ( t) = \ A \cos ( \omega t - \alpha ) = \ \mathop{\rm Re} [ Be ^ {i \omega t } ], $$

where $ x = x( t) $ is the value of the vibrating magnitude at the moment of time $ t $, $ | A | = | B | $ is the amplitude, $ \omega $ is the periodic (circular) frequency, and $ \alpha $ is the initial phase of the vibration. The duration of one complete vibration $ T = 2 \pi / \omega $ is called the period of the harmonic vibration, while $ \nu = 1/T $, which is the number of complete vibrations performed in unit time, is known as the frequency of the harmonic vibration ( $ \omega = 2 \pi \nu $). The period of the harmonic vibration is independent of its amplitude. The velocity, acceleration and all higher derivatives of the vibrating magnitude vary harmonically at the same frequency. A harmonic vibration is represented as an ellipse on the phase plane $ ( x, \dot{x} ) $. Owing to the dissipation of energy, perfect harmonic vibrations are not encountered in nature, but many processes are close to harmonic vibrations. These include small vibrations of mechanical systems with respect to their equilibrium position. The resulting frequencies (the so-called eigen frequencies) of the vibrations are independent of the initial conditions of motion and are determined by the nature of the vibrating system itself. For instance, small vibrations (under the effect of gravity) of a mathematical pendulum on a thread of length $ l $ are described by the differential equation

$$ ml \dot{x} dot = - mgx, $$

where $ g $ is the gravity acceleration and $ x( t) $ is the angle between the vertical and the thread of the pendulum. The general solution of this equation has the form $ x = A \cos ( \omega t - \alpha ) $, where the eigen frequency of the vibration, $ \omega = \sqrt g/l $, depends on $ g $ and $ l $ only, while the amplitude $ A $ and the phase $ \alpha $ are constants of integration, selected in accordance with the initial conditions.

Harmonic vibrations play an important part in the study of vibrations as a whole, since complicated periodic and almost-periodic varying magnitudes may be approximately represented, to any desired degree of accuracy, by a sum of harmonic vibrations. Mathematically this corresponds to the approximations of functions by trigonometric series and by Fourier integrals (cf. Fourier integral).

The classical Fourier series

$$ x ( t) = \ \sum _ {n = - \infty } ^ \infty a _ {n} e ^ {i nt } $$

of a complex-valued function $ x( t) $, defined on $ [ - \pi , \pi ] $, may be regarded as the expansion of $ x( t) $ into a sum of harmonic vibrations with integer frequencies $ n = 0, \pm 1, \pm 2 ,\dots $. The Fourier coefficient

$$ a _ {n} = { \frac{1}{2 \pi } } \int\limits _ {- \pi } ^ \pi x ( t) e ^ {- i n t } dt $$

determines the amplitude $ ( | a _ {n} | ) $ and the phase shift $ ( \mathop{\rm arg} a _ {n} ) $ of a harmonic vibration with frequency $ n $. The totality of all Fourier coefficients determines the spectrum of $ x( t) $ and shows the harmonic vibrations which are actually involved in $ x( t) $, as well as the amplitudes and the initial phases of these vibrations. Knowing the spectrum is equivalent to knowing the function $ x( t) $.

A function $ x( t) $ defined on $ ( - \infty , \infty ) $ can no longer be built from harmonic vibrations with integer frequencies. Its construction involves vibrations of all frequencies: The function $ x( t) $ can be represented by a Fourier integral:

$$ x ( t) = \int\limits _ {- \infty } ^ \infty a ( n) e ^ {i nt } dn ; $$

where

$$ a ( n) = { \frac{1}{2 \pi } } \int\limits _ {- \infty } ^ \infty x ( t) e ^ {- in t } dt $$

is the spectral density of $ x( t) $.

These representations of functions form the base of the Fourier method for solving various problems in the theory of differential and integral equations.

References

[1]	G.S. Gorelik, "Oscillations and waves" , Moscow-Leningrad (1950) (In Russian)

References

[a1]	J.W.S. Rayleigh, "The theory of sound" , 1 , Dover, reprint (1945)