Editing Oscillation (section)

==Damped oscillations==
{{Main|Harmonic oscillator}}
{{see also|Anti-vibration compound}}
[[File:Phase_portrait_of_damped_oscillator,_with_increasing_damping_strength.gif|thumb|Phase portrait of damped oscillator, with increasing damping strength.]]
All real-world oscillator systems are [[Thermodynamic reversibility|thermodynamically irreversible]].  This means there are dissipative processes such as [[friction]] or [[electrical resistance]] which continually convert some of the energy stored in the oscillator into heat in the environment.  This is called damping.  Thus, oscillations tend to decay with time unless there is some net source of energy into the system. The simplest description of this decay process can be illustrated by oscillation decay of the harmonic oscillator.

Damped oscillators are created when a resistive force is introduced, which is dependent on the first derivative of the position, or in this case velocity. The differential equation created by Newton's second law adds in this resistive force with an arbitrary constant {{mvar|b}}. This example assumes a linear dependence on velocity.
<math display="block">m\ddot{x} + b\dot{x} + kx = 0</math>

This equation can be rewritten as before:
<math display="block">\ddot{x} + 2 \beta \dot{x} + \omega_0^2x = 0,</math> 
where <math display="inline">2 \beta = \frac b m</math>.

This produces the general solution:
<math display="block">x(t) = e^{- \beta t} \left(C_1e^{\omega _1 t} + C_2 e^{- \omega_1t}\right),</math>
where <math display="inline">\omega_1 = \sqrt{\beta^2 - \omega_0^2}</math>.

The exponential term outside of the parenthesis is the [[Exponential decay|decay function]] and {{mvar|β}} is the damping coefficient. There are 3 categories of damped oscillators: under-damped, where {{math|''β'' < ''ω''<sub>0</sub>}}; over-damped, where {{math|''β'' > ''ω''<sub>0</sub>}}; and critically damped, where {{math|1=''β'' = ''ω''<sub>0</sub>}}.