Degree of damping

The degree of attenuation , also damping or Lehrsches attenuation (by Ernst teaching ), common symbols is, in the physics a measure of the damping of an oscillatory system. It is a quantity in the number dimension . It can be read from it how the system behaves after a stimulus . ${\displaystyle D}$

background

The differential equation of a linear damped oscillator can be given the following form, regardless of the physical background of the oscillation system:

${\displaystyle {\ddot {x}}+2D\omega _{0}{\dot {x}}+\omega _{0}^{2}x=0}$

There are:

• ${\displaystyle D}$: Degree of damping
• ${\displaystyle \omega _{0}}$: Natural angular frequency of the undamped system

Mechanical systems

For a spring / mass oscillator, Lehr's damping is calculated as follows:

${\displaystyle D={\frac {d}{2{\sqrt {km}}}}\ ={\frac {d\omega _{0}}{2k}}\ ={\frac {d}{2m\omega _{0}}}}$

There are:

${\displaystyle d}$: Damping constant

${\displaystyle k}$: Spring constant or spring stiffness

${\displaystyle m}$: Mass

The characteristic frequency corresponds to the natural frequency of the undamped system and is here . ${\displaystyle \omega _{0}={\sqrt {\frac {k}{m}}}}$

Based on the usage in English, the degree of damping can be understood as the ratio of the damping constant to the critical damping constant . This means ${\displaystyle d}$ ${\displaystyle d_{k}}$

${\displaystyle D={\frac {d}{d_{k}}}}$

The critical damping constant is the damping that is necessary to achieve the aperiodic limit case .

Electrical systems

The following applies to electrical oscillating circuits (see quality factor )

for the series resonant circuit:	for the parallel resonant circuit:
${\displaystyle D={\frac {R}{2L\omega _{0}}}={\frac {R}{2}}\cdot {\sqrt {\,{\frac {C}{L}}\,}}}$	${\displaystyle D={\frac {1}{2RC\omega _{0}}}={\frac {1}{2R}}\cdot {\sqrt {\,{\frac {L}{C}}\,}}}$

Are there

${\displaystyle R}$: Resistance

${\displaystyle C}$: Capacity

${\displaystyle L}$: Inductance

Analysis of stability

The degree of damping can be used to characterize the vibration behavior. For this one considers the solution of the characteristic polynomial of the differential equation:

${\displaystyle \lambda _{1,2}=-\omega _{0}(D\pm {\sqrt {D^{2}-1}})}$

A distinction is now made depending on the size of the degree of damping:

• ${\displaystyle D<0}$: unstable - upward swinging system
• ${\displaystyle D=0}$: undamped, borderline stable - continuous oscillation with constant amplitude
• ${\displaystyle 0<D<1}$: damped oscillation (case of weak damping)
• ${\displaystyle D=1}$: aperiodic borderline case - precisely no overshoot (case of critical damping)
• ${\displaystyle D>1}$: aperiodic solution - not oscillating (asymptotic approach to the center of oscillation for , creep case) ${\displaystyle t\rightarrow \infty }$

Other attenuation dimensions

Logarithmic decrement

The degree of damping describes the vibration behavior of an entire physical system. It is directly related to the logarithmic decrement via the equation: ${\displaystyle D}$ ${\displaystyle \Lambda }$

${\displaystyle D={\frac {\Lambda }{\sqrt {(2\pi )^{2}+\Lambda ^{2}}}}.}$

This quantity can also be found as a logarithmic attenuation measure in dB.

Attenuation level in acoustics

In the case of a plane wave, the attenuation measure with the symbol is the logarithmic ratio of the amplitudes of a field size (e.g. sound pressure) at two points lying one behind the other in the direction of sound propagation; (DIN 1320). ${\displaystyle a}$

Attenuation in electrical engineering

In electrical engineering, the damping behavior of resonant circuits is indicated by the quality factor . The relationship between the quality factor and the degree of damping applies: ${\displaystyle Q}$

${\displaystyle Q={\frac {1}{2D}}}$

literature

• Michael M. Rieländer: Real Lexicon of Acoustics. Erwin Bochinsky publishing house, Frankfurt am Main 1982, ISBN 3-920112-84-9