Damping (disambiguation): Difference between revisions - Wikipedia
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{{wiktionary}}
{{about|damped harmonic oscillators|detailed mathematical description of the harmonic oscillator including forcing and damping|Harmonic oscillator|damping in music|Damping (music)}}
'''[[Damping]]''' is an influence within or upon an oscillatory system that has the effect of reducing or preventing its [[oscillation]].
{{Classical mechanics |core}}
[[File:Damped spring.gif|right|frame|Underdamped spring–mass system]]
'''Damping''' is an influence within or upon an [[oscillator|oscillatory system]] that has the effect of reducing, restricting or preventing its oscillations. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples include [[Viscosity|viscous]] [[Drag (physics)|drag]] in mechanical systems, [[Electrical resistance and conductance|resistance]] in [[electronic oscillators]], and absorption and scattering of light in [[optical oscillator]]s. Damping not based on energy loss can be important in other oscillating systems such as those that occur in [[ecology|biological systems]].
'''Damping''' may also refer to:
The damping of a system can be described as being one of the following:
;Overdamped: The system returns ([[Exponential decay|exponentially decays]]) to equilibrium without oscillating.
;Critically damped: The system returns to equilibrium as quickly as possible without oscillating.
;Underdamped: The system oscillates (at reduced frequency compared to the ''undamped'' case) with the amplitude gradually decreasing to zero.
;Undamped: The system oscillates at its natural resonant frequency (''ω''<sub>o</sub>) without experiencing decay of its amplitude.
For example, consider a door that uses a spring to close the door once open. This can lead to any of the above types of damping depending on the strength of the damping. If the door is ''undamped'' it will swing back and forth forever at a particular resonant frequency. If it is ''underdamped'' it will swing back and forth with decreasing size of the swing until it comes to a stop. If it is ''critically damped'' then it will return to closed as quickly as possible without oscillating. Finally, if it is ''overdamped'' it will return to closed without oscillating but more slowly depending on how overdamped it is. Different levels of damping are desired for different types of systems.
== Linear damping ==
A particularly mathematically useful type of damping is linear damping. Linear damping occurs when a potentially oscillatory variable is damped by an influence that opposes changes in it, in direct proportion to the instantaneous rate of change, velocity or [[derivative|time derivative]], of the variable itself. In engineering applications it is often desirable to [[linearization|linearize]] non-linear drag forces. This may be done by finding an equivalent work coefficient in the case of harmonic forcing. In non-harmonic cases, restrictions on the speed may lead to accurate linearization.
In [[physics]] and [[engineering]], damping may be [[Mathematical model|mathematically modeled]] as a [[force]] synchronous with the [[velocity]] of the object but opposite in direction to it. If such force is also proportional to the velocity, as for a simple mechanical [[viscous]] damper ([[dashpot]]), the force <math>F</math> may be related to the velocity <math>v</math> by
:<math> F = -cv \, ,</math>
where ''c'' is the ''damping coefficient'', given in units of newton-seconds per meter.
This force may be used as an approximation to the [[friction]] caused by [[drag (physics)|drag]] and may be realized, for instance, using a [[dashpot]]. (This device uses the viscous drag of a fluid, such as oil, to provide a resistance that is related linearly to velocity.) Even when [[friction]] is related to <math>v^2</math>, if the velocity is restricted to a small range, then this non-linear effect may be small. In such a situation, a linearized friction coefficient <math>c_{lin}</math> may be determined which produces little error.
When including a restoring force (such as due to a spring) that is proportional to the displacement <math>x</math> and in the opposite direction, and by setting the sum of these two forces equal to the mass of the object times its acceleration creates a second-order [[Ordinary differential equation|differential equation]] whose terms can be rearranged into the following form:
:<math> \frac{d^2x}{dt^2} + 2\zeta\omega_0\frac{dx}{dt} + \omega_0^2 x = 0,</math>
where ''ω''<sub>0</sub> is the undamped [[angular frequency]] of the oscillator and ''ζ'' is a constant called the [[damping ratio]]. This equation is valid for many different oscillating systems, but with different formulas for the damping ratio and the undamped angular frequency.
The value of the damping ratio ''ζ'' determines the behavior of the system such that ''ζ'' = 1 corresponds to being critically damped with larger values being overdamped and smaller values being underdamped. If ''ζ'' = 0, the system is undamped.
=== Example: mass–spring–damper ===
[[File:Коливальна система із пружиною та демпфером.png|thumb|Mass attached to a spring and damper.]]
An ideal mass–[[Hooke's law|spring]]–damper system with mass ''m'', [[spring constant]] ''k'', and [[viscous]] damper of damping coefficient ''c'' is subject to an oscillatory force
: <math>F_\mathrm{s} = - k x \,</math>
and a damping force
:<math>F_\mathrm{d} = - c v = - c \frac{dx}{dt} = - c \dot{x}.</math>
The values can be in any consistent system of units; for example, in [[SI unit]]s, ''m'' in [[kilograms]], ''k'' in [[Newton (unit)|newtons]] per meter, and ''c'' in [[newton-second]]s per meter or [[kilograms]] per [[second]].
Treating the mass as a [[free body]] and applying [[Newton's laws of motion#Newton's second law|Newton's second law]], the total force ''F''<sub>tot</sub> on the body is
:<math>F_\mathrm{tot} = ma = m \frac{d^2x}{dt^2} = m \ddot{x}.</math>
where ''a'' is the [[acceleration]] of the mass and ''x'' is the [[displacement (vector)|displacement]] of the mass relative to a fixed point of reference.
Since ''F''<sub>tot</sub> = ''F''<sub>s</sub> + ''F''<sub>d</sub>,
:<math>m \ddot{x} = -kx + -c\dot{x}.</math>
This differential equation may be rearranged into
: <math>\ddot{x} + { c \over m} \dot{x} + {k \over m} x = 0.\,</math>
The following parameters are then defined:
: <math>\omega_0 = \sqrt{ k \over m }</math>
: <math>\zeta = { c \over 2 \sqrt{m k} }.</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>\gamma</math> ([[gamma]]) is, in general, a [[complex number]].
Substituting this assumed solution back into the differential equation gives
:<math>\gamma^2 + 2 \zeta \omega_0 \gamma + \omega_0^2 = 0 \, ,</math>
which is the [[Linear differential equation#Homogeneous equations with constant coefficients|characteristic equation]].
Solving the characteristic equation will give two roots
<math> \gamma_{\pm}=-\zeta \omega_0 \pm \omega_0 \sqrt{\zeta ^2 - 1}.</math>
The solution to the differential equation is thus<ref>[http://mathworld.wolfram.com/DampedSimpleHarmonicMotionOverdamping.html MathWorld--A Wolfram Web Resource]</ref>
: <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>
=== System behavior ===
[[File:Damping 1.svg|thumb|300px|Time dependence of the system behavior on the value of the damping ratio ''ζ'', for undamped ''(blue)'', under-damped ''(green)'', critically damped ''(red)'', and over-damped ''(cyan)'' cases, for zero-velocity initial condition.]]
[[File:Resonance.PNG|thumb|300px|Steady state variation of amplitude with frequency and damping of a driven [[simple harmonic oscillator]].<ref>
{{cite book
| author = Katsuhiko Ogata
| title = System Dynamics
| edition = 4th
| publisher = University of Minnesota
| year = 2005
| page = 617
}}</ref><ref>
{{cite book
| title = Optics, 3E
| author = [[Ajoy Ghatak]]
| edition = 3rd
| publisher = Tata McGraw-Hill
| year = 2005
| isbn = 978-0-07-058583-6
| page = 6.10
| url = https://books.google.com/books?id=jStDc2LmU5IC&pg=PT97
}}</ref>]]
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 ''γ'' has one real solution, two real solutions, or two complex conjugate solutions.
====Critical damping (''ζ'' = 1) ====
When {{nowrap|''ζ'' {{=}} 1}}, there is a double root ''γ'' (defined above), which is real. The system is said to be ''critically damped''. A critically damped system converges to zero as fast as possible without oscillating (although overshoot can occur). 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, with only one root ''γ'', there is in addition to the solution {{nowrap|''x''(''t'') {{=}} ''e<sup>γt</sup>''}} a solution {{nowrap|''x''(''t'') {{=}} ''te<sup>γt</sup>''}}:<ref>Weisstein, Eric W. "Critically Damped Simple Harmonic Motion." From MathWorld--A Wolfram Web Resource. [http://mathworld.wolfram.com/CriticallyDampedSimpleHarmonicMotion.html]</ref>
: <math> x(t) = (A+Bt)\,e^{-\omega_0 t}, </math>
where <math>A</math> and <math>B</math> 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 (''ζ'' > 1)====<!-- This section is linked from [[Over-damping]] and [[Overdamping]] -->
When ''ζ'' > 1, the system is ''over-damped'', and there are two different real roots. An over-damped door-closer takes longer to close than a critically damped door does.
The solution to the motion equation is:<ref>{{MathWorld | title = Damped Simple Harmonic Motion--Overdamping. | urlname = DampedSimpleHarmonicMotionOverdamping }}</ref>
: <math> x(t) = Ae^{\gamma_+ t} + Be^{\gamma_- t}, </math>
where <math>A</math> and <math>B</math> 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 (0 ≤ ''ζ'' < 1)====
{{anchor|Under-damping}}
Finally, when 0 < ''ζ'' < 1, ''γ'' is [[complex number|complex]], and the system is ''under-damped''. In this situation, the system will oscillate at the natural damped frequency ''ω''<sub>d</sub>, which is a function of the natural frequency and the damping ratio. To continue the analogy, an underdamped door closer would close quickly, but would hit the door frame with significant velocity, or would oscillate in the case of a swinging door.
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 ''damped frequency'' or ''ringing frequency'' of the system,<ref>
{{cite book
| title = Electrical Engineering License Review
| edition = 8th
| author = Lincoln D. Jones
| publisher = Dearborn Trade Publishing
| year = 2003
| isbn = 978-0-7931-8529-0
| page = 6‑15 <!-- a page, not a range! -->
| url = https://books.google.com/books?id=9QXpWQqlwNQC&pg=PT174
}}</ref>
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>
This "damped frequency" is not to be confused with the ''damped resonant frequency'' or ''peak frequency'' ''ω''<sub>peak</sub>.<ref>
{{cite book
| title = Principles of engineering mechanics
| edition =
| author = Millard F. Beatty
| publisher = Birkhäuser
| year = 2006
| isbn = 978-0-387-23704-6
| page = 167
| url = https://books.google.com/books?id=k4GgsitYDt4C&pg=PA167
}}</ref>
This is the frequency at which a moderately underdamped (''ζ'' < 1/{{radic|2}}) simple 2nd-order harmonic oscillator has its maximum gain (or peak [[Transmissibility (vibration)|transmissibility]]) when driven by a [[Sine wave|sinusoidal]] input. The frequency at which this peak occurs is given by:
:<math>\omega_\textrm{peak} = \omega_0\sqrt{1 - 2\zeta^2}.</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 ==
{{unreferenced section|date=February 2016}}
Viscous damping models, although widely used, are not the only damping models. A wide range of models can be found in specialized literature. One is 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, a measure of the fraction of energy lost in each cycle of the vibration.
: <math>
\eta = E_d / ( \pi k X^2 ),
</math>
where ''E<sub>d</sub>'' is the energy lost and X is the amplitude of the cycle.
Although requiring [[complex analysis]] to solve the equation, this model reproduces the real behavior of many vibrating structures more closely than the viscous model.
A more general model that also requires complex analysis, the fractional model not only includes both the viscous and hysteretic models but also allows for intermediate cases (useful for some polymers):
: <math>
m \ddot{x} + A \frac{d^r x}{dt^r} i + k x = 0,
</math>
where ''r'' is any number, usually between 0 (for hysteretic) and 1 (for viscous), and ''A'' is a general damping (''h'' for hysteretic and ''c'' for viscous) coefficient.
{{see also|Friction|Drag (physics)|l2=Drag}}
===Nonlinear damping===
Nonlinear passive damping offers important advantages compared to purely linear designs.<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> Nonlinear damping using an odd function such as cubic damping, allows the user to damp the resonance without increasing the energy in the frequency response tail and so overcomes several limitations of a purely linear design.
== {{Anchor|misusage}} Errors in popular usage ==
It has become common in popular English, especially in [[Star Trek]],<ref>[http://www.chakoteya.net/startrek/31.htm The Star Trek Transcripts - Metamorphosis]</ref><ref>[http://www.chakoteya.net/movies/movie7.html The Movie Transcripts - Star Trek Generations]</ref> to substitute the word ''dampening'' when the concept of ''damping'' is intended. Defined as to make damp or to stifle,<ref>[http://www.collinsdictionary.com/dictionary/english/dampening dampening - Collins English Dictionary - Complete & Unabridged 10th Edition]</ref> dampening can be correctly used to describe depressing the intensity of an emotion, but should not be used to describe the reduction in amplitude of a force, a harmonic oscillation, or similar physical processes or phenomena. For such phenomena, "damping" is the correct word.<ref>
{{cite book
| title = Fundamentals of Acoustics
| edition = 3, illustrated
| author = Lawrence E. Kinsler
| publisher = Wiley
| year = 1982
| isbn = 0471029335
| page = 7
| url = https://books.google.com/books?id=ibTvAAAAMAAJ
}}</ref><ref>
{{cite book
| title = Mechanical Vibrations
| author = J. P. Den Hartog
| publisher = Courier Dover
| year = 1985
| isbn = 0486647854
| page = 7
| url = https://books.google.com/books?id=-Pu5YlgY4QsC
}}</ref><ref>
{{cite book
| title = Fundamentals of Vibrations
| author = Leonard Meirovitch
| publisher = McGraw-Hill Higher Education
| year = 2002
| isbn = 0072881801
| pages = 25–27
| url = https://books.google.com/books?id=dQ4EAAAACAAJ
}}</ref>
== See also ==
{{colbegin||20em}}
* [[Audio system measurements]]
* [[Control theory]]
* [[Coulomb damping]]
* [[Damping factor]]
* [[Damping ratio(music)]]
* [[HarmonicDamping balancertorque]]
* [[HarmonicDamping oscillatorcapacitor]]
* [[ImpulseDamping excitation techniquematrix]]
* [[Damping off]]
* [[Inerter (mechanical networks)]]
* [[Oscillator]]
* [[Particle damping]]
* [[Resonance]]
* [[RLC circuit]]
* [[Simple harmonic motion]]
* [[Thermoelastic damping]]
* [[Thrust damping]]
* [[Tuned mass damper]]
* [[Vehicle suspension]]
* [[Vibration]]
* [[Vibration control]]
{{colend}}
== References ==
{{reflist|30em}}
==Books==
* [[Vadim Komkov|Komkov, Vadim]] (1972) Optimal control theory for the damping of vibrations of simple elastic systems. Lecture Notes in Mathematics, Vol. 253. Springer-Verlag, Berlin-New York.
== External links ==
{{Wiktionary}}
* [http://www.sengpielaudio.com/calculator-bridging.htm Calculation of the matching attenuation,the damping factor, and the damping of bridging]
* [http://vibrationdata.wordpress.com/category/damping/ Damping Matlab scripts]
{{disambig}}
[[Category:Mechanical vibrations]]
[[Category:Ordinary differential equations]]
[[Category:Control theory]]