# 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 .

## background

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

There are:

- : Degree of damping
- : Natural angular frequency of the undamped system

### Mechanical systems

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

There are:

- : Damping constant
- : Spring constant or spring stiffness
- : Mass

The characteristic frequency corresponds to the natural frequency of the undamped system and is here .

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

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: |

Are there

- : Resistance
- : Capacity
- : 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:

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

- : unstable - upward swinging system
- : undamped, borderline stable - continuous oscillation with constant amplitude
- : damped oscillation (case of weak damping)
- : aperiodic borderline case - precisely no overshoot (case of critical damping)
- : aperiodic solution - not oscillating (asymptotic approach to the center of oscillation for , creep case)

## 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:

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).

### 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:

## literature

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