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 : 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