Angular frequency

The angular frequency or angular frequency is a physical quantity theory of vibrations. As symbols of the Greek small letter omega is used. It is a measure of how fast a vibration. In contrast to frequency, but does not indicate the number of periods of vibration, based on a period of time, but the phase angle of the swept cycle per time period. Since a period of oscillation corresponds to a phase angle, the angular frequency of the frequency difference by a factor of:

Wherein the period of the oscillation. The unit of angular frequency. Unlike the frequency of this unit is not designated at the angular frequency as the " Hertz".

Pointer model

Harmonic oscillations can be represented by the rotation of a pointer, the length of which corresponds to the amplitude of the vibration. The Momentanauslenkung is the projection of the pointer on one of the coordinate axes. When using the complex number plane for the display of the pointer corresponds to - depending on the definition of - either the real part or the imaginary part of Momentanauslenkung.

The angular frequency is the rate of change of the phase angle of the rotating pointer ( see picture ). In adaptation to the unity of the angular frequency of the angle should this be specified in radians.

The link model is applicable to all types of vibrations (mechanical, electrical, etc.) and signals applicable. Since an oscillation period of a full rotation of the pointer corresponds to and is the full angle, is the angular frequency of a harmonic wave is always the times its frequency. Often the indication of the angular frequency versus frequency is preferred because many formulas of the theory of oscillations can be more compact display due to the occurrence of trigonometric functions whose period is, by definition, using the angular frequency: eg with a simple cosine oscillation: instead.

In the case of temporary non-constant angular frequency, the term is used instantaneous angular frequency.

Use in the theory of oscillations

A harmonic oscillation can generally be described as a function of angular frequency:

They can, as is usual in electrical engineering, are represented by the real and imaginary parts of a rotating complex phasor with constant angular velocity in the Gaussian plane as a function of angular frequency, and time. The time-dependent angle of the complex vector is referred to as phase angle.

The relationship with the sine and cosine results from Euler's formula.

Characteristic angular frequency and natural angular frequency

Oscillatory systems are described by the characteristic angular frequency and the natural angular frequency. An undamped free vibrating system resonates with its characteristic angular frequency, a damped system without external excitation oscillates at its natural angular frequency. The natural angular frequency of a damped system is always smaller than the characteristic angular frequency. The characteristic angular frequency is called as a mechanical in undamped natural angular frequency.

For the example of an electrical resonant circuit is considered with the resistance, inductance and capacitance for the characteristic angular frequency:

A spring pendant with the spring stiffness, the mass and the damping constants applies to the characteristic angular frequency:

And the decay constant or the natural angular frequency:

For more examples see the torsion pendulum, water pendulum, pendulum.

Complex angular frequency

From the complex phasor representation of a harmonic oscillation

Obtained with the traditional approach

The generalization of the complex angular frequency with the real part and the angular frequency. By the complex angular frequency, not only a constant harmonic oscillation may be shown by, but with a damped oscillation and excited oscillation. A classic application of the complex angular frequency is the advanced symbolic method of alternating current technology.

A damped oscillation S can be represented as a complex phasor with constant complex angular frequency as follows:

This is the natural angular frequency of the oscillatory system and is equal to the negative value of the decay constant: (see the previous section ).

In the Laplace transform the complex angular frequency has a more general meaning as a variable in the image area of the transformation to the representation of arbitrary time functions and transfer functions in the complex frequency plane ( " s-plane ").

Relationship to the angular velocity

Often the term " angular frequency " is introduced by a mechanical analogy: If you have a point of a rotating body (or a rotating vector) projected perpendicular to the axis of rotation to a level, you get the picture of a harmonic ( sinusoidal ) oscillation. The angular frequency of vibration resulting from these projections, this has the same value as the angular velocity of the rotating body. However, this projection is only the mechanical illustration of an abstract concept: harmonic ( ie, sinusoidal ) oscillation is shown in the complex plane by the rotation of a complex phasor. Through this abstraction, the term angular frequency of vibrations of any kind is (electrical, mechanical, etc.) applicable and has no direct relation to rotating bodies. The angular frequency of the abstract describes rate of change of the phase angle in the complex plane, whereas the angular velocity of describe the change of a physical angle to a physical body by the change in time.