Angular frequency

Physical size
Surname	Angular frequency, angular frequency
Formula symbol	${\displaystyle \omega }$
Derived from	frequency
Size and unit system unit dimension SI s −1 T −1

Radians for angles: The angle that cuts the length of the circle radius from the circumference is 1 radian . The full angle is therefore radians. ${\displaystyle 2\pi }$

The angular frequency or angular frequency is a physical quantity in vibration theory . The Greek letter (small omega ) is used as a symbol . It is a measure of how fast an oscillation takes place. In contrast to the frequency , however, it does not indicate the number of oscillation periods in relation to a time span , but the swept phase angle of the oscillation per time span. Since one oscillation period corresponds to a phase angle of , the angular frequency differs from the frequency by a factor : ${\displaystyle \omega }$ ${\displaystyle f}$ ${\displaystyle 2\pi }$ ${\displaystyle 2\pi }$

${\displaystyle \omega =2\pi f={\frac {2\pi }{T}}}$ ,

where is the period of the oscillation. The unit of the angular frequency is . In contrast to frequency, this unit is not referred to as hertz for angular frequency . ${\displaystyle T}$ ${\displaystyle 1/\mathrm {s} }$

Pointer model

Vector representation of a harmonic oscillation in the complex plane ( using the example of an alternating voltage ) with the time-dependent argument . ${\displaystyle {\underline {u}}}$ ${\displaystyle \varphi =\omega t+\varphi _{0}}$

Harmonic oscillations can be represented by rotating a pointer whose length corresponds to the amplitude of the oscillation. The current deflection is the projection of the pointer onto one of the coordinate axes. If the complex number plane is used to represent the pointer , either the real part or the imaginary part corresponds to the instantaneous deflection, depending on the definition.

The angular frequency is the rate of change of the phase angle of the rotating pointer (see adjacent picture). In adaptation to the unit of the angular frequency, the angle should be given in radians. ${\displaystyle \omega }$ ${\displaystyle \varphi }$

${\displaystyle \omega ={\frac {{\text{d}}\varphi }{{\text{d}}t}}}$

The pointer model is applicable to all types of vibrations (mechanical, electrical, etc.) and signals. Since one oscillation period corresponds to a full turn of the pointer and the full angle is, the angular frequency of a harmonic oscillation is always times its frequency. Often the specification of the angular frequency is preferred to the frequency, since many formulas of the oscillation theory can be represented more compactly with the help of the angular frequency due to the occurrence of trigonometric functions whose period is by definition : z. B. for a simple cosine oscillation: instead of . ${\displaystyle 2\pi }$ ${\displaystyle 2\pi }$ ${\displaystyle 2\pi }$ ${\displaystyle y={\hat {y}}\cdot \cos(\omega t)}$ ${\displaystyle y={\hat {y}}\cdot \cos(2\pi f\,t)}$

In the case of angular frequencies that are not constant over time, the term instantaneous angular frequency is also used.

Use in vibration theory

A harmonic oscillation can generally be described as a function of the angular frequency : ${\displaystyle \omega }$

${\displaystyle x(t)=x_{0}\,\sin \left(\omega t+\varphi _{0}\right)}$

As is common in electrical engineering, it can be represented by the real and imaginary part of a complex pointer rotating at constant angular velocity in the Gaussian plane of numbers as a function of the angular frequency and time. The time-dependent angle of the complex vector is called the phase angle . ${\displaystyle {\underline {x}}}$ ${\displaystyle \varphi (t)=\omega t+\varphi _{0}}$

${\displaystyle {\underline {x}}(t)=x_{0}\,e^{i(\omega t+\varphi _{0})}=x_{0}\,(\cos(\omega t+\varphi _{0})+i\sin(\omega t+\varphi _{0}))}$

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

Characteristic angular frequency and natural angular frequency

Systems capable of oscillation are described by the characteristic angular frequency and the natural angular frequency . An undamped free oscillating system oscillates with its characteristic angular frequency , a damped system without external excitation oscillates with 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 also referred to in mechanics as the undamped natural angular frequency . ${\displaystyle \omega _{0}}$ ${\displaystyle \omega _{d}}$

For the example of an electrical oscillating circuit, the following applies with the resistance , the inductance and the capacitance for the characteristic frequency: ${\displaystyle R}$ ${\displaystyle L}$ ${\displaystyle C}$

${\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}}}$

For a spring pendulum with the spring stiffness and the mass, the following applies for the characteristic frequency: ${\displaystyle c}$ ${\displaystyle m}$

${\displaystyle \omega _{0}={\sqrt {\frac {c}{m}}}}$

and with the decay constant or for the natural angular frequency: ${\displaystyle \delta =R/{(2L)}}$ ${\displaystyle \delta =d/{(2m)}}$

${\displaystyle \omega _{d}={\sqrt {\omega _{0}^{2}-\delta ^{2}}}}$.

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

Complex angular frequency

From the complex pointer representation of a harmonic oscillation

${\displaystyle {\underline {x}}(t)=x_{0}\,e^{\mathrm {i} \omega t}}$

results with the usual approach

${\displaystyle {\underline {x}}(t)=x_{0}\,e^{st}}$

the generalization to the complex angular frequency with the real part and the angular frequency . Due to the complex angular frequency , not only a constant harmonic oscillation can be represented, but also a damped oscillation and an excited oscillation . A classic application of the complex angular frequency is the extended symbolic method of alternating current technology . ${\displaystyle s=\sigma +\mathrm {i} \,\omega }$ ${\displaystyle \sigma }$ ${\displaystyle \omega }$ ${\displaystyle s}$ ${\displaystyle \sigma =0}$ ${\displaystyle \sigma <0}$ ${\displaystyle \sigma >0}$

A damped oscillation can be represented as a complex vector with the constant complex angular frequency s as follows:

${\displaystyle {\underline {x}}(t)=x_{0}\,e^{st}=x_{0}\,e^{(\sigma +\mathrm {i} \omega _{d})t}=x_{0}\,e^{\sigma t}e^{\mathrm {i} \omega _{d}t}=x_{0}\,e^{\sigma t}(\cos(\omega _{d}t)+\mathrm {i} \sin(\omega _{d}t))}$

Here is the natural angular frequency of the oscillatable system and is equal to the negative value of the decay constant: (see the previous section). ${\displaystyle \omega _{d}}$ ${\displaystyle \sigma }$ ${\displaystyle \sigma =-\delta }$

In the Laplace transformation , the complex angular frequency has a more general meaning as a variable in the image area of ​​the transformation to represent any time functions and transfer functions in the complex frequency plane ("s-plane"). ${\displaystyle s=\sigma +\mathrm {i} \omega }$ ${\displaystyle F(s)}$

Relationship to angular velocity

Frequently, the term "angular frequency" is inserted by a mechanical analogy: If a point of a rotating body (or a rotating vector) perpendicular to the axis of rotation to a plane projected , one obtains the image of a harmonic (sinusoidal) oscillation. The angular frequency of the oscillation that results from this projection has the same numerical value as the angular velocity of the rotating body. However, this projection is only the mechanical illustration of an abstract concept: Harmonic (i.e. sinusoidal) vibrations are represented in the complex plane by the rotation of a complex vector. Due to this abstraction, the term angular frequency can be applied to vibrations of any kind (electrical, mechanical, etc.) and has no direct reference to rotating bodies. The angular frequency describes the abstract rate of change of the phase angle in the complex plane, while the angular velocity describes the change in a physical angle on a physical body per change in time.

Web links

Wiktionary: Radial frequency  - explanations of meanings, word origins, synonyms, translations

Individual evidence

1. ↑ DIN 1301-2 Units, Commonly Used Parts and Multiples
2. ↑ Detlef Kamke, Wilhelm Walcher: Physics for medical professionals . Springer DE, 1994, ISBN 3-322-80144-6 , pp. 43 . ( limited preview in the Google book search)
   Klaus Federn: Balancing technology Volume 1 . Springer DE, 2011, ISBN 3-642-17237-7 , pp. 104 . ( limited preview in Google Book search)
3. ↑ Eberhard Brommundt, Delf Sachau: Schwingungslehre: with machine dynamics . Springer, 2007 ( limited preview in Google Book Search).
4. ↑ The harmonic oscillation , math online
5. ↑ Wolf-Ewald Büttner: Fundamentals of Electrical Engineering, Volume 2 . 2nd Edition. Oldenbourg, 2009, ISBN 978-3-486-58981-8 , pp. 215 ( limited preview in Google Book search).
6. ^ Manfred Precht, Karl Voit, Roland Kraft: Mathematics 2 for non-mathematicians . Oldenbourg Verlag, 2005, ISBN 3-486-57775-1 , p. 69 ( limited preview in Google Book search). Douglas C. Giancoli: Physics: Upper School . Pearson Deutschland GmbH, 2010, ISBN 3-86894-903-8 , p.
    170 ( limited preview in Google Book search). Jürgen Eichler: Physics: for engineering studies - concise with almost 300 sample tasks . Springer DE, 2011, ISBN 3-8348-9942-9 , pp.
    112 ( limited preview in Google Book search).