Damping
495563
223079697
2008-07-02T13:53:28Z
Ignat99
2572681
[[Image:Damped spring.gif|right|frame|A damped [[spring-mass system]].]]
'''Damping''' is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of [[oscillation]]s of an oscillatory system.
== Definition ==
In [[physics]] and [[engineering]], damping may be [[Mathematical model|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
: <math>
\bold{F} = -c \bold{v}
</math>
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]].
This force is an (raw) approximation to the [[friction]] caused by [[drag (physics)|drag]].
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 ==
[[Image:Mass-Spring-Damper.png|thumb|300px|A mass attached to a spring and damper. The damping coefficient, usually ''c'', is represented by ''B'' in this case. The ''F'' in the diagram denotes an external force, which this example does not include.]]
An ideal mass-spring-damper system with mass ''m'' (in [[kilograms]]), spring constant ''k'' (in [[newton]]s per [[meter]]) and viscous damper of damping coefficient ''c'' (in [[newton-second]]s per meter) can be described with the following formula:
: <math>
F_\mathrm{s} = - k x
</math>
: <math>
F_\mathrm{d} = - c v = - c \dot{x} = - c \frac{dx}{dt}
</math>
Treating the mass as a [[free body]] and applying [[Newton's laws of motion#Newton's second law: Law of Motion|Newton's second law]], we have:
: <math>
\sum F = ma = m \ddot{x} = m \frac{d^2x}{dt^2}
</math>
where ''a'' is the [[acceleration]] (in meters per second squared) of the mass and ''x'' is the [[displacement (vector)|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 [[second]]s):
: <math>
m \ddot{x} + c \dot{x} + k x = 0.\,
</math>
Rearranging, we have
: <math>
\ddot{x} + { c \over m} \dot{x} + {k \over m} x = 0.\,
</math>
Next, to simplify the equation, we define the following parameters:
: <math>
\omega_0 = \sqrt{ k \over m }
</math>
and
: <math>
\zeta = { c \over 2 \sqrt{k m} }.
</math>
The first parameter, ω<sub>0</sub>, is called the (undamped) [[resonance|natural frequency]] of the system .
The second parameter, ζ, is called the ''[[damping ratio]]''. The
natural frequency represents an [[angular frequency]], expressed in [[radian]]s per second. The damping ratio is a [[dimensionless quantity]].
The differential equation now becomes
: <math>
\ddot{x} + 2 \zeta \omega_0 \dot{x} + \omega_0^2 x = 0.\,
</math>
Continuing, we can solve the equation by assuming a solution x such that:
: <math>
x = e^{\gamma t}\,
</math>
where the [[parameter]] <math>\scriptstyle \gamma</math> is, in general, a [[complex number]].
Substituting this assumed solution back into the differential equation, we obtain
: <math>
\gamma^2 + 2 \zeta \omega_0 \gamma + \omega_0^2 = 0.\,
</math>
Solving for γ, we find:
: <math>
\gamma = \omega_0( - \zeta \pm \sqrt{\zeta^2 - 1}).
</math>
=== System behavior ===
[[Image:Damping.svg|thumb|300px|Dependence of the system behavior on the value of the damping ratio <math>\zeta</math>.]]
The behavior of the system depends on the relative values of the two fundamental parameters, the natural frequency ω<sub>0</sub> and the damping ratio ζ.
In particular, the qualitative behavior of the system depends crucially on whether
the [[quadratic equation]] for <math>\scriptstyle\gamma</math> has one real solution, two real solutions, or
two complex conjugate solutions.
==== Critical damping ====
When '''ζ = 1''', <math>\scriptstyle\gamma</math> (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<ref>Weisstein, Eric W. "Damped Simple Harmonic Motion--Critical Damping." From MathWorld--A Wolfram Web Resource. [http://mathworld.wolfram.com/DampedSimpleHarmonicMotionCriticalDamping.html]</ref>:
: <math>
x(t) = (A+Bt)\,e^{-\omega_0 t} \,
</math>
where ''A'' and ''B'' are determined by the initial conditions of the system (usually the initial position and velocity of the mass):
: <math>
A = x(0) \,
</math>
: <math>
B = \dot{x}(0)+\omega_0x(0) \,
</math>
==== Over-damping ====
When '''ζ > 1''', <math>\scriptstyle\gamma</math> 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<ref>Weisstein, Eric W. "Damped Simple Harmonic Motion--Overdamping." From MathWorld--A Wolfram Web Resource. [http://mathworld.wolfram.com/DampedSimpleHarmonicMotionOverdamping.html]</ref>:
<!-- The equations in the reference page are wrong! the typo has already been reported to Wolfram Mathworld -->
: <math>
x(t) = Ae^{\gamma_+ t} + Be^{\gamma_- t}
</math>
where ''A'' and ''B'' are determined by the initial conditions of the system:
: <math>
A = x(0)+\frac{\gamma_+x(0)-\dot{x}(0)}{\gamma_--\gamma_+}
</math>
: <math>
B = -\frac{\gamma_+x(0)-\dot{x}(0)}{\gamma_--\gamma_+}.
</math>
==== Under-damping ====
Finally, when '''0 ≤ ζ < 1''', <math>\scriptstyle\gamma</math> is [[complex number|complex]], and the system is '''''under-damped'''''. In this situation, the system will oscillate at the natural damped frequency <math> \scriptstyle\omega_\mathrm{d}</math>, which is a function of the natural frequency and the damping ratio.
In this case, the solution can be generally written as<ref>Weisstein, Eric W. "Damped Simple Harmonic Motion--Underdamping." From MathWorld--A Wolfram Web Resource. [http://mathworld.wolfram.com/DampedSimpleHarmonicMotionUnderdamping.html]</ref>:
: <math>
x (t) = e^{- \zeta \omega_0 t} (A \cos\,(\omega_\mathrm{d}\,t) + B \sin\,(\omega_\mathrm{d}\,t ))\,
</math>
where
: <math>
\omega_\mathrm{d} = \omega_0 \sqrt{1 - \zeta^2 }\,
</math>
represents the ''natural damped frequency'' of the system, and ''A'' and ''B'' are again determined by the initial conditions of the system:
: <math>
A = x(0)\,
</math>
: <math>
B = \frac{1}{\omega_\mathrm{d}}(\zeta\omega_0x(0)+\dot{x}(0)).\,
</math>
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:
: <math>
m \ddot{x} + h x i + k x = 0
</math>
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:
: <math>
m \ddot{x} + k ( 1 + i \eta ) x = 0
</math>
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.
{{see also|Friction|Drag (physics)|l2=Drag}}
== In Music ==
===Guitar===
On [[guitar]], damping (also referred to as '''choking''') is a technique where, shortly after playing the [[string (music)|strings]], the sound is reduced by pressing the right hand palm against the strings, '''right hand damping''' (including [[Palm mute|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 (music)|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 [[fret]]ted string producing the desired note, ie the adjacent strings are scratched. See also: [[Bang]]/[[Skank (guitar)|Skank]]/[[Cheka (artist)|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. <ref>
Bolton, Ross (2001). ''Funk Guitar: The Essential Guide''. ISBN 0-634-01168-5.</ref>
===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 [[saron (instrument)|saron]]s and [[gendér]]s. On instruments that are played with a single [[mallet]], the left hand is used to damp the previously hit note when a new [[note (music)|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 ==
<div style="-moz-column-count:3; column-count:3;">
* [[Damping ratio]]
* [[Damping factor]]
* [[RLC circuit]]
* [[Oscillator]]
* [[Harmonic oscillator]]
* [[Simple harmonic motion]]
* [[Resonance]]
* [[Impulse excitation technique]]
* [[Audio system measurements]]
* [[Tuned mass damper]]
* [[Vehicle suspension]]
* [[Vibration]]
* [[Vibration control]]
* [[Thrust damping]]
</div>
== References ==
{{reflist}}
== External links ==
{{Wiktionary}}
* [http://www.sengpielaudio.com/calculator-bridging.htm Calculation of the matching attenuation,the damping factor, and the damping of bridging]
[[Category:Mechanical vibrations]]
[[Category:Ordinary differential equations]]
[[Category:Electronics terms]]
[[Category:Control theory]]
* [http://www.stationcrafts.net/physics A-level Physics experiment on the subject of Damped Harmonic Motion with solution curve graphs]
[[ar:تخميد]]
[[cs:Tlumené kmitání]]
[[de:Dämpfung]]
[[fr:Amortissement physique]]
[[it:Ammortizzamento]]
[[pl:Tłumienie]]
[[ru:Затухающие колебания]]
[[sv:Dämpning]]
[[zh:阻尼]]