Editing Harmonic oscillator (section)

== Driven harmonic oscillators ==
<!-- Driving DAB entry and [[Stiff equation]] link here, pls do not change -->

Driven harmonic oscillators are damped oscillators further affected by an externally applied force ''F''(''t'').

[[Newton's second law]] takes the form
<math display="block">F(t) - kx - c\frac{\mathrm{d}x}{\mathrm{d}t}=m\frac{\mathrm{d}^2x}{\mathrm{d}t^2}. </math>

It is usually 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 = \frac{F(t)}{m}. </math>

This equation can be solved exactly for any driving force, using the solutions ''z''(''t'') that satisfy the unforced equation
<math display="block"> \frac{\mathrm{d}^2z}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}z}{\mathrm{d}t} + \omega_0^2 z = 0,</math>

and which can be expressed as damped sinusoidal oscillations:
<math display="block">z(t) = A e^{-\zeta \omega_0 t} \sin \left( \sqrt{1 - \zeta^2} \omega_0 t + \varphi \right), </math>
in the case where {{math|''ζ'' ≤ 1}}. The amplitude ''A'' and phase ''φ'' determine the behavior needed to match the initial conditions.

===Step input===
{{See also|Step response}}

In the case {{math|''ζ'' < 1}} and a unit step input with&nbsp;{{math|1=''x''(0) = 0}}:
<math display="block"> \frac{F(t)}{m} = \begin{cases} \omega _0^2 & t \geq 0 \\ 0 & t < 0 \end{cases}</math>
the solution is
<math display="block"> x(t) = 1 - e^{-\zeta \omega_0 t} \frac{\sin \left( \sqrt{1 - \zeta^2} \omega_0 t + \varphi \right)}{\sin(\varphi)},</math>

with phase ''φ'' given by

<math display="block">\cos \varphi = \zeta.</math>

The time an oscillator needs to adapt to changed external conditions is of the order {{math|1=''τ'' = 1/(''ζω''<sub>0</sub>)}}. In physics, the adaptation is called [[relaxation (physics)|relaxation]], and ''τ'' is called the relaxation time.

In electrical engineering, a multiple of ''τ'' is called the ''settling time'', i.e. the time necessary to ensure the signal is within a fixed departure from final value, typically within 10%. The term ''overshoot'' refers to the extent the response maximum exceeds final value, and ''undershoot'' refers to the extent the response falls below final value for times following the response maximum.

===Sinusoidal driving force===
[[File:Mplwp resonance zeta envelope.svg|thumb|300px|Steady-state variation of amplitude with relative frequency <math>\omega/\omega_0</math> and damping <math>\zeta</math> of a driven harmonic oscillator. This plot is also called the harmonic oscillator spectrum or motional spectrum.]]
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{{cite book
 | title = Optics, 3E
 | author = Ajoy Ghatak
 | author-link = 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>
-->

In the case of a sinusoidal driving force:
<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 = \frac{1}{m} F_0 \sin(\omega t),</math>
where <math>F_0</math> is the driving amplitude, and <math>\omega</math> is the driving [[frequency]] for a sinusoidal driving mechanism. This type of system appears in [[alternating current|AC]]-driven [[RLC circuit]]s ([[Electrical resistance|resistor]]–[[inductor]]–[[capacitor]]) and driven spring systems having internal mechanical resistance or external [[air resistance]].

The general solution is a sum of a [[Transient (oscillation)|transient]] solution that depends on initial conditions, and a [[steady state]] that is independent of initial conditions and depends only on the driving amplitude <math>F_0</math>, driving frequency <math>\omega</math>, undamped angular frequency <math>\omega_0</math>, and the damping ratio <math>\zeta</math>.

The steady-state solution is proportional to the driving force with an induced phase change <math>\varphi</math>:
<math display="block"> x(t) = \frac{F_0}{m Z_m \omega} \sin(\omega t + \varphi),</math>
where
<math display="block" qid=Q6421317> Z_m = \sqrt{\left(2\omega_0\zeta\right)^2 + \frac{1}{\omega^2} (\omega_0^2 - \omega^2)^2}</math>
is the absolute value of the [[Mechanical impedance|impedance]] or [[linear response function]], and
<math display="block"> \varphi = \arctan\left(\frac{2\omega \omega_0\zeta}{\omega^2 - \omega_0^2} \right) + n\pi</math>

is the [[phase (waves)|phase]] of the oscillation relative to the driving force. The phase value is usually taken to be between −180° and 0 (that is, it represents a phase lag, for both positive and negative values of the arctan argument).

For a particular driving frequency called the [[resonance]], or resonant frequency <math display="inline">\omega_r = \omega_0 \sqrt{1 - 2\zeta^2}</math>, the amplitude (for a given <math>F_0</math>) is maximal. This resonance effect only occurs when <math>\zeta < 1 / \sqrt{2}</math>, i.e. for significantly underdamped systems. For strongly underdamped systems the value of the amplitude can become quite large near the resonant frequency.

The transient solutions are the same as the unforced (<math>F_0 = 0</math>) damped harmonic oscillator and represent the system's response to other events that occurred previously.