Editing Harmonic oscillator (section)

== Damped harmonic oscillator ==
{{main|Mass-spring-damper model|damping}}
[[File:Damping 1.svg|thumb|Dependence of the system behavior on the value of the damping ratio ''ζ'']]
[[File:Phase_portrait_of_damped_oscillator,_with_increasing_damping_strength.gif|thumb|Phase portrait of damped oscillator, with increasing damping strength.]]
[[File:Oscillatory motion acceleration.ogv|thumb|Video clip demonstrating a damped harmonic oscillator consisting of a dynamics cart between two springs. An [[accelerometer]] on top of the cart shows the magnitude and direction of the acceleration.]]

In real oscillators, friction, or damping, slows the motion of the system. Due to frictional force, the velocity decreases in proportion to the acting frictional force. While in a simple undriven harmonic oscillator the only force acting on the mass is the restoring force, in a damped harmonic oscillator there is in addition a frictional force which is always in a direction to oppose the motion. In many vibrating systems the frictional force ''F''<sub>f</sub> can be modeled as being proportional to the velocity ''v'' of the object: {{math|1=''F''<sub>f</sub> = −''cv''}}, where ''c'' is called the ''[[viscous damping]] coefficient''.

The balance of forces ([[Newton's second law]]) for damped harmonic oscillators is then<ref>{{harvtxt|Fowles|Cassiday|1986|p=86}}</ref><ref>{{harvtxt|Kreyszig|1972|p=65}}</ref><ref>{{harvtxt|Tipler|1998|pp=369,389}}</ref>
<math display="block"> F = - kx - c\frac{\mathrm{d}x}{\mathrm{d}t} = m \frac{\mathrm{d}^2x}{\mathrm{d}t^2},</math>
which can be rewritten into the form
<math display="block"> \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = 0, </math>
where
* <math display="inline">\omega_0 = \sqrt{\frac k m}</math> is called the "undamped [[angular frequency]] of the oscillator",
* <math qid="Q1127660" display="inline">\zeta = \frac{c}{2\sqrt{mk}}</math> is called the "[[Damping#Damping ratio|damping ratio]]".

[[Image:Step response for two-pole feedback amplifier.PNG|thumb|[[Step response]] of a damped harmonic oscillator; curves are plotted for three values of {{nowrap|1=''μ'' = ''ω''<sub>1</sub> = ''ω''<sub>0</sub>{{radic|1 − ''ζ''<sup>2</sup>}}}}. Time is in units of the decay time {{nowrap|1=''τ'' = 1/(''ζω''<sub>0</sub>)}}.]]

The value of the damping ratio ''ζ'' critically determines the behavior of the system. A damped harmonic oscillator can be:
* ''Overdamped'' (''ζ'' > 1): The system returns ([[exponential decay|exponentially decays]]) to steady state without oscillating.  Larger values of the damping ratio ''ζ'' return to equilibrium more slowly.
* ''Critically damped'' (''ζ'' = 1):  The system returns to steady state as quickly as possible without oscillating (although overshoot can occur if the initial velocity is nonzero).  This is often desired for the damping of systems such as doors.
* ''Underdamped'' (''ζ'' < 1): The system oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero. The [[angular frequency]] of the underdamped harmonic oscillator is given by <math display="inline">\omega_1 = \omega_0\sqrt{1 - \zeta^2},</math> the [[exponential decay]] of the underdamped harmonic oscillator is given by <math>\lambda = \omega_0\zeta.</math>

The [[Q factor]] of a damped oscillator is defined as 
<math display="block">Q = 2\pi \times \frac{\text{energy stored}}{\text{energy lost per cycle}}.</math>

''Q'' is related to the damping ratio by <math display="inline">Q = \frac{1}{2\zeta}.</math>