# User Contributed Dictionary

### Verb

damping- present participle of damp

### Noun

- The reduction in the magnitude of oscillations by the dissipation of energy
- The stabilization of a physical system by reducing oscillation

# Extensive Definition

Damping is any effect, either deliberately
engendered or inherent to a system, that tends to reduce the
amplitude of oscillations of an
oscillatory system.

## Definition

In physics and engineering, damping may be
mathematically
modelled as a force
synchronous with the velocity of the object but
opposite in direction to it. Thus, for a simple mechanical damper,
the force F may be related to the velocity v by

\bold = -c \bold

where c is the viscous damping coefficient, given
in units of newton-seconds per meter.

This relationship is perfectly analogous to
electrical
resistance. See Ohm's
law.

In playing stringed instruments such as guitar or violin, damping is the quieting
or abrupt silencing of the strings after they have been sounded, by
pressing with the edge of the palm, or other parts of the hand such
as the fingers on one or more strings near the bridge of the
instrument. The strings themselves can be modelled as a continuum
of infinitesimally small mass-spring-damper systems where the
damping constant is much smaller than the resonance frequency,
creating damped oscillations (see below). See also Vibrating
string.

## Example: mass-spring-damper

An ideal mass-spring-damper system with mass m
(in kilograms), spring
constant k (in newtons
per meter) and viscous
damper of damping coefficient c (in newton-seconds
per meter) can be described with the following formula:

F_\mathrm = - k x

F_\mathrm = - c v = - c \dot = - c \frac

Treating the mass as a free body and
applying
Newton's second law, we have:

\sum F = ma = m \ddot = m \frac

where a is the acceleration (in meters per
second squared) of the mass and x is the displacement
(in meters) of the mass relative to a fixed point of
reference.

### Differential equation

The above equations combine to form the equation
of motion, a second-order differential
equation for displacement x as a function of time t (in
seconds):

m \ddot + c \dot + k x = 0.\,

Rearranging, we have

\ddot + \dot + x = 0.\,

Next, to simplify the equation, we define the
following parameters:

\omega_0 = \sqrt

and

\zeta = .

The first parameter, ω0, is called the (undamped)
natural frequency of
the system . The second parameter, ζ, is called the damping
ratio. The natural frequency represents an angular
frequency, expressed in radians per second. The damping
ratio is a dimensionless
quantity.

The differential equation now becomes

\ddot + 2 \zeta \omega_0 \dot + \omega_0^2 x =
0.\,

Continuing, we can solve the equation by assuming
a solution x such that:

x = e^\,

where the parameter \scriptstyle \gamma
is, in general, a complex
number.

Substituting this assumed solution back into the
differential equation, we obtain

\gamma^2 + 2 \zeta \omega_0 \gamma + \omega_0^2 =
0.\,

Solving for γ, we find:

\gamma = \omega_0( - \zeta \pm \sqrt).

### System behavior

The behavior of the system depends on the
relative values of the two fundamental parameters, the natural
frequency ω0 and the damping ratio ζ. In particular, the
qualitative behavior of the system depends crucially on whether the
quadratic
equation for \scriptstyle\gamma has one real solution, two real
solutions, or two complex conjugate solutions.

#### Critical damping

When ζ = 1, \scriptstyle\gamma (defined above) is
real, the system is said to be critically damped. A critically
damped system converges to zero faster than any other without
oscillating. An example of critical damping is the door-closer seen
on many hinged doors in public buildings. The recoil mechanisms in
most guns are also critically damped so that they return to their
original position, after the recoil due to firing, in the least
possible time.

In this case, the solution simplifies to:

x(t) = (A+Bt)\,e^ \,

where A and B are determined by the initial
conditions of the system (usually the initial position and velocity
of the mass):

A = x(0) \,

B = \dot(0)+\omega_0x(0) \,

#### Over-damping

When ζ > 1, \scriptstyle\gamma is still real,
but now the system is said to be over-damped. An over-damped
door-closer will take longer to close than a critically damped door
would.

The solution to the motion equation is:

x(t) = Ae^ + Be^

where A and B are determined by the initial
conditions of the system:

A = x(0)+\frac

B = -\frac.

#### Under-damping

Finally, when 0 ≤ ζ \scriptstyle\gamma is
complex,
and the system is under-damped''. In this situation, the system
will oscillate at the natural damped frequency
\scriptstyle\omega_\mathrm, which is a function of the natural
frequency and the damping ratio.

In this case, the solution can be generally
written as:

x (t) = e^ (A \cos\,(\omega_\mathrm\,t) + B
\sin\,(\omega_\mathrm\,t ))\,

where

\omega_\mathrm = \omega_0 \sqrt\,

represents the natural damped frequency of the
system, and A and B are again determined by the initial conditions
of the system:

A = x(0)\,

B = \frac(\zeta\omega_0x(0)+\dot(0)).\,

For an under-damped system, the value
of ζ can be found by examining the logarithm of the ratio
of succeeding amplitudes of a system. This is called the logarithmic
decrement.

## Alternative models

Viscous damping models, although widely used, are
not the only damping models. A wide range of models can be found in
specialized literature, but one of them should be referred here:
the so called "hysteretic damping model" or "structural damping
model".

When a metal beam is vibrating, the internal
damping can be better described by a force proportional to the
displacement but in phase with the velocity. In such case, the
differential
equation that describes the free movement of a
single-degree-of-freedom system becomes:

m \ddot + h x i + k x = 0

where h is the hysteretic damping coefficient and
i denotes the imaginary
unit; the presence of i is required to synchronize the damping
force to the velocity ( xi being in phase with the velocity). This
equation is more often written as:

m \ddot + k ( 1 + i \eta ) x = 0

where η is the hysteretic damping ratio, that is,
the fraction of energy lost in each cycle of the vibration.

Although requiring complex
analysis to solve the equation, this model reproduces the real
behaviour of many vibrating structures more closely than the
viscous model.

## In Music

### Guitar

On guitar, damping (also referred to as choking) is a technique where, shortly after playing the strings, the sound is reduced by pressing the right hand palm against the strings, right hand damping (including Palm muting), or relaxing the left hand fingers' pressure on the strings, left hand damping (or Left-hand muting). Scratching is where the strings are played while damped, ie, the strings are damped before playing. The term presumably refers to the clunky sound produced. In funk music this is often done over a sixteenth note pattern with occasional sixteenths undamped.Floating is the technique where a chord is
sustained past a sixteenth note rather than that note being
scratched, the term referring to the manner in which the right hand
"floats" over the strings rather than continuing to scratch.

Skanking is where a note is isolated by left hand
damping of the two strings adjacent to the fully fretted string producing the
desired note, ie the adjacent strings are scratched. See also:
Bang/Skank/Cheka.
The technique is extremely popular among Reggae, Ska, and Rocksteady
guitarists, who uses it with virtually every riddim they play on. It is a
classical element of this style of music.

Damping is possible on other string instruments
by halting the vibration of the strings using the left hand,
similar to on a guitar.

### Piano

On a piano, damping is controlled by the sustain pedal, with the strings being damped unless the pedal is pressed.### Gamelan

Damping is also important in most percussion instruments in the gamelan, especially the sarons and gendérs. On instruments that are played with a single mallet, the left hand is used to damp the previously hit note when a new note is played. On the gendér, which is played with mallets in both hands, the keys must be damped by the same hand, and it requires practice to master the technique.## See also

## References

## External links

damping in Arabic: تخميد

damping in Czech: Tlumené kmitání

damping in German: Dämpfung

damping in French: Amortissement physique

damping in Polish: Tłumienie

damping in Swedish: Dämpning

damping in Chinese: 阻尼