Editing Harmonic oscillator

{{Short description|Physical system that responds to a restoring force inversely proportional to displacement}}
{{About|the harmonic oscillator in classical mechanics|its uses in [[quantum mechanics]]|quantum harmonic oscillator}}
{{Use American English|date = February 2019}}
{{Classical mechanics|Core}}

In [[classical mechanics]], a '''harmonic oscillator''' is a system that, when displaced from its [[Mechanical equilibrium|equilibrium]] position, experiences a [[restoring force]] ''F'' [[Proportionality (mathematics)|proportional]] to the [[Displacement (geometry)|displacement]] ''x'':
<math display="block" qid=Q170282> \vec F = -k \vec x, </math>
where ''k'' is a [[positive number|positive]] [[coefficient|constant]].

The harmonic oscillator model is important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as [[Clock|clocks]] and radio circuits.

If ''F'' is the only force acting on the system, the system is called a '''simple harmonic oscillator''', and it undergoes [[simple harmonic motion]]: [[sinusoidal]] [[oscillation]]s about the [[equilibrium point]], with a constant [[amplitude]] and a constant [[frequency]] (which does not depend on the amplitude).

If a frictional force ([[Damping ratio|damping]]) proportional to the [[velocity]] is also present, the harmonic oscillator is described as a '''damped oscillator'''. Depending on the friction coefficient, the system can:
* Oscillate with a frequency lower than in the [[Damping ratio|undamped]] case, and an [[amplitude]] decreasing with time ([[Damping ratio|underdamped]] oscillator).
* Decay to the equilibrium position, without oscillations ([[Damping ratio|overdamped]] oscillator).

The [[Boundary value problem|boundary solution]] between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called [[critically damped]].

If an external time-dependent force is present, the harmonic oscillator is described as a ''driven oscillator''.

Mechanical examples include [[pendulum]]s (with [[Small-angle approximation#Motion of a pendulum|small angles of displacement]]), masses connected to [[spring (device)|springs]], and [[acoustics|acoustical systems]]. Other [[#Equivalent systems|analogous systems]] include electrical harmonic oscillators such as [[RLC circuit]]s. They are the source of virtually all sinusoidal vibrations and waves.

== Simple harmonic oscillator ==
{{main|Simple harmonic motion}}
{{multiple image
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| image1   = Animated-mass-spring-faster.gif
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A simple harmonic oscillator is an oscillator that is neither driven nor [[Damping ratio|damped]]. It consists of a mass ''m'', which experiences a single force ''F'', which pulls the mass in the direction of the point {{math|1=''x'' = 0}} and depends only on the position ''x'' of the mass and a constant ''k''. Balance of forces ([[Newton's second law]]) for the system is
<math display="block">F = m a = m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} = m\ddot{x} = -k x. </math>

Solving this [[differential equation]], we find that the motion is described by the function
<math display="block" qid=Q3299367> x(t) = A \sin(\omega t + \varphi), </math>
where
<math display="block" qid=Q834020>\omega = \sqrt{\frac k m}.</math>

The motion is [[Periodic function|periodic]], repeating itself in a [[sine wave|sinusoidal]] fashion with constant amplitude ''A''. In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its [[Frequency|period]] <math>T = 2\pi/\omega</math>, the time for a single oscillation or its frequency <math>f=1/T</math>, the number of cycles per unit time. The position at a given time ''t'' also depends on the [[phase (waves)|phase]] ''φ'', which determines the starting point on the sine wave. The period and frequency are determined by the size of the mass ''m'' and the force constant ''k'', while the amplitude and phase are determined by the starting position and [[velocity]].

The velocity and [[acceleration]] of a simple harmonic oscillator oscillate with the same frequency as the position, but with shifted phases.  The velocity is maximal for zero displacement, while the acceleration is in the direction opposite to the displacement.

The potential energy stored in a simple harmonic oscillator at position ''x'' is
<math display="block" qid="Q891408">U = \tfrac 1 2 kx^2.</math>

== 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>

== 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
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 | edition = 3rd
 | publisher = Tata McGraw-Hill
 | year = 2005
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 | url = https://books.google.com/books?id=jStDc2LmU5IC&pg=PT97 
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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.

== Parametric oscillators ==
{{main|Parametric oscillator}}
A [[parametric oscillator]] is a driven harmonic oscillator in which the drive energy is provided by varying the parameters of the oscillator, such as the damping or restoring force.
A familiar example of parametric oscillation is "pumping" on a playground [[swing (seat)|swing]].<ref name=Case>{{cite web |title=Two ways of driving a child's swing |url=http://www.grinnell.edu/academic/physics/faculty/case/swing/ |first=William |last=Case |access-date=27 November 2011 |url-status=dead |archive-url=https://web.archive.org/web/20111209230414/http://www.grinnell.edu/academic/physics/faculty/case/swing |archive-date=9 December 2011 }}</ref><ref name=Case96>{{Cite journal | last1 = Case | first1 = W. B. | title = The pumping of a swing from the standing position | doi = 10.1119/1.18209 | journal = American Journal of Physics | volume = 64 | issue = 3 | pages = 215–220 | year = 1996 |bibcode = 1996AmJPh..64..215C }}</ref><ref name=Roura>{{cite journal |last1=Roura |first1=P. |last2=Gonzalez |first2=J.A. |year=2010 |title=Towards a more realistic description of swing pumping due to the exchange of angular momentum |journal=European Journal of Physics |volume=31 |issue=5 |pages=1195–1207 |doi=10.1088/0143-0807/31/5/020 |bibcode = 2010EJPh...31.1195R |s2cid=122086250 }}</ref>
A person on a moving swing can increase the amplitude of the swing's oscillations without any external drive force (pushes) being applied, by changing the moment of inertia of the swing by rocking back and forth ("pumping") or alternately standing and squatting, in rhythm with the swing's oscillations.  The varying of the parameters drives the system.  Examples of parameters that may be varied are its resonance frequency <math>\omega</math> and damping <math>\beta</math>.

Parametric oscillators are used in many applications.  The classical [[varactor]] parametric oscillator oscillates when the diode's capacitance is varied periodically.  The circuit that varies the diode's capacitance is called the "pump" or "driver".   In microwave electronics, [[waveguide (electromagnetism)|waveguide]]/[[Yttrium aluminum garnet|YAG]] based parametric oscillators operate in the same fashion. The designer varies a parameter periodically to induce oscillations.

Parametric oscillators have been developed as low-noise amplifiers, especially in the radio and microwave frequency range.  Thermal noise is minimal, since a reactance (not a resistance) is varied.  Another common use is frequency conversion, e.g., conversion from audio to radio frequencies.  For example, the [[Optical parametric oscillator]] converts an input [[laser]] wave into two output waves of lower frequency (<math>\omega_s, \omega_i</math>).

Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies. Parametric excitation differs from forcing, since the action appears as a time varying modification on a system parameter. This effect is different from regular resonance because it exhibits the [[instability]] phenomenon.

==Universal oscillator equation==
The equation 
<math display="block">\frac{\mathrm{d}^2q}{\mathrm{d} \tau^2} + 2 \zeta \frac{\mathrm{d}q}{\mathrm{d}\tau} + q = 0</math>
is known as the '''universal oscillator equation''', since all second-order linear oscillatory systems can be reduced to this form.{{citation needed|date=October 2018}} This is done through [[nondimensionalization]].

If the forcing function is {{math|1=''f''(''t'') = cos(''ωt'') = cos(''ωt<sub>c</sub>τ'') = cos(''ωτ'')}}, where {{math|1=''ω'' = ''ωt''<sub>''c''</sub>}}, the equation becomes
<math display="block">\frac{\mathrm{d}^2q}{\mathrm{d} \tau^2} + 2 \zeta \frac{\mathrm{d}q}{\mathrm{d}\tau} + q = \cos(\omega \tau).</math>

The solution to this differential equation contains two parts: the "transient" and the "steady-state".

=== Transient solution ===
The solution based on solving the [[ordinary differential equation]] is for arbitrary constants ''c''<sub>1</sub> and ''c''<sub>2</sub>

<math display="block">q_t (\tau) = \begin{cases}
 e^{-\zeta\tau} \left( c_1 e^{\tau \sqrt{\zeta^2 - 1}} + c_2 e^{- \tau \sqrt{\zeta^2 - 1}} \right) & \zeta > 1 \text{ (overdamping)} \\
 e^{-\zeta\tau} (c_1+c_2 \tau) = e^{-\tau}(c_1+c_2 \tau) & \zeta = 1 \text{ (critical damping)} \\
 e^{-\zeta \tau} \left[ c_1 \cos \left(\sqrt{1-\zeta^2} \tau\right) + c_2 \sin\left(\sqrt{1-\zeta^2} \tau\right) \right] & \zeta < 1 \text{ (underdamping)}
\end{cases}</math>

The transient solution is independent of the forcing function.

=== Steady-state solution ===
Apply the "[[complex analysis|complex variables]] method" by solving the [[auxiliary equation]] below and then finding the real part of its solution:
<math display="block">\frac{\mathrm{d}^2 q}{\mathrm{d}\tau^2} + 2 \zeta \frac{\mathrm{d}q}{\mathrm{d}\tau} + q = \cos(\omega \tau) + i\sin(\omega \tau) = e^{ i \omega \tau}.</math>

Supposing the solution is of the form
<math display="block">q_s(\tau) = A e^{i (\omega \tau + \varphi) }. </math>

Its derivatives from zeroth to second order are
<math display="block">q_s = A e^{i (\omega \tau + \varphi) }, \quad
\frac{\mathrm{d}q_s}{\mathrm{d} \tau} = i \omega A e^{i (\omega \tau + \varphi) }, \quad
\frac{\mathrm{d}^2 q_s}{\mathrm{d} \tau^2} = -\omega^2 A e^{i (\omega \tau + \varphi) } .</math>

Substituting these quantities into the differential equation gives
<math display="block">-\omega^2 A e^{i (\omega \tau + \varphi)} + 2 \zeta i \omega A e^{i(\omega \tau + \varphi)} + A e^{i(\omega \tau + \varphi)} = (-\omega^2 A + 2 \zeta i \omega A + A) e^{i (\omega \tau + \varphi)} = e^{i \omega \tau}.</math>

Dividing by the exponential term on the left results in
<math display="block">-\omega^2 A + 2 \zeta i \omega A + A = e^{-i \varphi} = \cos\varphi - i \sin\varphi.</math>

Equating the real and imaginary parts results in two independent equations
<math display="block">A (1 - \omega^2) = \cos\varphi, \quad 2 \zeta \omega A = -\sin\varphi.</math>

==== Amplitude part ====
[[Image:Harmonic oscillator gain.svg|thumb|[[Bode plot]] of the frequency response of an ideal harmonic oscillator]]
Squaring both equations and adding them together gives
<math display="block">\left. \begin{aligned}
 A^2 (1-\omega^2)^2 &= \cos^2\varphi \\
 (2 \zeta \omega A)^2 &= \sin^2\varphi
\end{aligned} \right\}
\Rightarrow A^2[(1 - \omega^2)^2 + (2 \zeta \omega)^2] = 1.</math>

Therefore,
<math display="block">A = A(\zeta, \omega) = \sgn \left( \frac{-\sin\varphi}{2 \zeta \omega} \right) \frac{1}{\sqrt{(1 - \omega^2)^2 + (2 \zeta \omega)^2}}.</math>

Compare this result with the theory section on [[resonance]], as well as the "magnitude part" of the [[RLC circuit]]. This amplitude function is particularly important in the analysis and understanding of the [[frequency response]] of second-order systems.

==== Phase part ====
To solve for {{math|''φ''}}, divide both equations to get
<math display="block">\tan\varphi = -\frac{2 \zeta \omega}{1 - \omega^2} = \frac{2 \zeta \omega}{\omega^2 - 1}~~ \implies ~~ \varphi \equiv \varphi(\zeta, \omega) = \arctan \left( \frac{2 \zeta \omega}{\omega^2 - 1} \right ) + n\pi.</math>

This phase function is particularly important in the analysis and understanding of the [[frequency response]] of second-order systems.

=== Full solution ===
Combining the amplitude and phase portions results in the steady-state solution
<math display="block">q_s(\tau) = A(\zeta,\omega) \cos(\omega \tau + \varphi(\zeta, \omega)) = A\cos(\omega \tau + \varphi).</math>

The solution of original universal oscillator equation is a [[Superposition principle|superposition]] (sum) of the transient and steady-state solutions:
<math display="block">q(\tau) = q_t(\tau) + q_s(\tau).</math>

==Equivalent systems==
{{main|System equivalence}}
Harmonic oscillators occurring in a number of areas of engineering are equivalent in the sense that their mathematical models are identical (see [[#Universal oscillator equation|universal oscillator equation]] above).  Below is a table showing analogous quantities in four harmonic oscillator systems in mechanics and electronics.  If analogous parameters on the same line in the table are given numerically equal values, the behavior of the oscillators{{snd}}their output waveform, resonant frequency, damping factor, etc.{{snd}}are the same.

{|class="wikitable"
! Translational mechanical
! Rotational mechanical
![[RLC circuit#Series circuit|Series RLC circuit]]
![[RLC circuit#Parallel circuit|Parallel RLC circuit]]
|-
| Position <math>x</math>
| Angle <math>\theta</math>
| [[Charge (physics)|Charge]] <math>q</math>
| [[Flux linkage]] <math>\varphi</math>
|-
| [[Velocity]] <math>\frac{\mathrm{d}x}{\mathrm{d}t}</math>
| [[Angular velocity]] <math>\frac{\mathrm{d}\theta}{\mathrm{d}t}</math>
| [[Electric current|Current]] <math>\frac{\mathrm{d}q}{\mathrm{d}t}</math>
| [[Voltage]] <math>\frac{\mathrm{d}\varphi}{\mathrm{d}t}</math>
|-
| [[Mass]] <math>m</math>
| [[Moment of inertia]] <math>I</math>
| [[Inductance]] <math>L</math>
| [[Capacitance]] <math>C</math> 
|-
|[[Momentum]] <math>p</math>
|[[Angular momentum]] <math>L</math>
|[[Flux linkage]] <math>\varphi</math>
|[[Electric charge|Charge]] <math>q</math>
|-
| [[Hooke's law|Spring constant]] <math>k</math>
| [[Torsion spring#Torsion coefficient|Torsion constant]] <math>\mu</math>
| [[Elastance]] <math>1/C</math>
| [[Magnetic reluctance]] <math>1/L</math>
|-
| [[Damping ratio|Damping]] <math>c</math>
| [[Torsion spring#Motion of torsion balances and pendulums|Rotational friction]] <math>\Gamma</math>
| [[Electrical resistance|Resistance]] <math>R</math>
| [[Electrical conductance|Conductance]] <math>G = 1/R</math> 
|-
| Drive [[force]] <math>F(t)</math>
| Drive [[torque]] <math>\tau(t)</math>
| [[Voltage]] <math>v</math>
| [[Electric current|Current]] <math>i</math> 
|-
|colspan="4" align="center"| Undamped [[Resonance|resonant frequency]] <math>f_n</math>:
|-
| <math>\frac{1}{2\pi}\sqrt{\frac{k}{m}}</math>
| <math>\frac{1}{2\pi}\sqrt{\frac{\mu}{I}}</math>
| <math>\frac{1}{2\pi}\sqrt{\frac{1}{LC}}</math>
| <math>\frac{1}{2\pi}\sqrt{\frac{1}{LC}}</math>
|-
|colspan="4" align="center"| [[Damping ratio]] <math>\zeta</math>:
|-
| <math>\frac{c}{2}\sqrt{\frac{1}{km}}</math>
| <math>\frac{\Gamma}{2}\sqrt{\frac{1}{I\mu}}</math>
| <math>\frac{R}{2}\sqrt{\frac{C}{L}}</math>
| <math>\frac{G}{2}\sqrt{\frac{L}{C}}</math>
|-
|colspan="4" align="center"| Differential equation:
|-
| <math>m \ddot x + c \dot x + kx = F</math>
| <math>I \ddot\theta + \Gamma \dot\theta + \mu\theta = \tau</math>
| <math>L \ddot q + R \dot q + q/C = v</math>
| <math>C \ddot\varphi + G \dot\varphi + \varphi/L = i</math>
|}

== Application to a conservative force ==
The problem of the simple harmonic oscillator occurs frequently in physics, because a mass at equilibrium under the influence of any [[conservative force]], in the limit of small motions, behaves as a simple harmonic oscillator.

A conservative force is one that is associated with a [[potential energy]].  The potential-energy function of a harmonic oscillator is
<math display="block">V(x) = \tfrac{1}{2} k x^2.</math>

Given an arbitrary potential-energy function <math>V(x)</math>, one can do a [[Taylor series|Taylor expansion]] in terms of <math>x</math> around an energy minimum (<math>x = x_0</math>) to model the behavior of small perturbations from equilibrium.

<math display="block">V(x) = V(x_0) + V'(x_0) \cdot (x - x_0) + \tfrac{1}{2} V''(x_0) \cdot (x - x_0)^2 + O(x - x_0)^3.</math>

Because <math>V(x_0)</math> is a minimum, the first derivative evaluated at <math>x_0</math> must be zero, so the linear term drops out:
<math display="block">V(x) = V(x_0) + \tfrac{1}{2} V''(x_0) \cdot (x - x_0)^2 + O(x - x_0)^3.</math>

The [[constant term]] {{math|''V''(''x''<sub>0</sub>)}} is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved:
<math display="block">V(x) \approx \tfrac{1}{2} V''(0) \cdot x^2 = \tfrac{1}{2} k x^2.</math>

Thus, given an arbitrary potential-energy function <math>V(x)</math> with a non-vanishing second derivative, one can use the solution to the simple harmonic oscillator to provide an approximate solution for small perturbations around the equilibrium point.

==Examples==

===Simple pendulum===
[[Image:Simple pendulum height.svg|thumb|A [[simple pendulum]] exhibits approximately simple harmonic motion under the conditions of no damping and small amplitude.]]

Assuming no damping, the differential equation governing a simple pendulum of length <math>l</math>, where <math>g</math> is the local [[Gravitational acceleration|acceleration of gravity]], is
<math display="block" qid=Q20702>\frac{d^2\theta}{dt^2} + \frac{g}{l}\sin\theta = 0.</math>

If the maximal displacement of the pendulum is small, we can use the approximation <math>\sin\theta \approx \theta</math> and instead consider the equation
<math display="block">\frac{d^2\theta}{dt^2} + \frac{g}{l}\theta = 0.</math>

The general solution to this differential equation is
<math display="block">\theta(t) = A \cos\left(\sqrt{\frac{g}{l}} t + \varphi \right),</math>
where <math>A</math> and <math>\varphi</math> are constants that depend on the initial conditions.
Using as initial conditions <math>\theta(0) = \theta_0</math> and <math>\dot{\theta}(0) = 0</math>, the solution is given by
<math display="block">\theta(t) = \theta_0 \cos\left(\sqrt{\frac{g}{l}} t\right),</math>
where <math>\theta_0</math> is the largest angle attained by the pendulum (that is, <math>\theta_0</math> is the amplitude of the pendulum).  The [[Sine|period]], the time for one complete oscillation, is given by the expression
<math display="block" qid=Q3382125>\tau = 2\pi \sqrt\frac{l}{g} = \frac{2\pi}{\omega},</math>
which is a good approximation of the actual period when <math>\theta_0</math> is small. Notice that in this approximation the period <math>\tau</math> is independent of the amplitude <math>\theta_0</math>. In the above equation, <math>\omega</math> represents the angular frequency.

===Spring/mass system===
[[Image:Harmonic oscillator.svg|thumb|Spring–mass system in equilibrium (A), compressed (B) and stretched (C) states]]

When a spring is stretched or compressed by a mass, the spring develops a restoring force. [[Hooke's law]] gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length:
<math display="block">F(t) = -kx(t),</math>
where ''F'' is the force, ''k'' is the spring constant, and ''x'' is the displacement of the mass with respect to the equilibrium position. The minus sign in the equation indicates that the force exerted by the spring always acts in a direction that is opposite to the displacement (i.e. the force always acts towards the zero position), and so prevents the mass from flying off to infinity.

By using either force balance or an energy method, it can be readily shown that the motion of this system is given by the following differential equation:
<math display="block"> F(t) = -kx(t) = m \frac{\mathrm{d}^2}{\mathrm{d} t^2} x(t) = ma, </math>
the latter being [[Newton's laws of motion#Newton's second law|Newton's second law of motion]].

If the initial displacement is ''A'', and there is no initial velocity, the solution of this equation is given by
<math display="block"> x(t) = A \cos \left( \sqrt{\frac{k}{m}} t \right).</math>

Given an ideal massless spring, <math>m</math> is the mass on the end of the spring. If the spring itself has mass, its [[Effective mass (spring-mass system)|effective mass]] must be included in <math>m</math>.

====Energy variation in the spring–damping system====
In terms of energy, all systems have two types of energy: [[potential energy]] and [[kinetic energy]]. When a spring is stretched or compressed, it stores elastic potential energy, which is then transferred into kinetic energy. The potential energy within a spring is determined by the equation <math display="inline"> U = \frac{1}{2}kx^2. </math>

When the spring is stretched or compressed, kinetic energy of the mass gets converted into potential energy of the spring. By [[conservation of energy]], assuming the datum is defined at the equilibrium position, when the spring reaches its maximal potential energy, the kinetic energy of the mass is zero. When the spring is released, it tries to return to equilibrium, and all its potential energy converts to kinetic energy of the mass.

==Definition of terms==

{| class='wikitable'
|-
! scope="col" | Symbol !! scope="col" | Definition!! scope="col" | Dimensions !! scope="col" | SI units 
|-
|<math>a</math>||Acceleration of mass||<math>\mathsf{LT^{-2}}</math>|| m/s<sup>2</sup> 
|-
|<math>A</math>||Peak amplitude of oscillation||<math>\mathsf{L}</math>|| m
|-
|<math>c</math>||Viscous damping coefficient||<math>\mathsf{MT^{-1}}</math>|| N·s/m
|-
|<math>f</math>||Frequency||<math>\mathsf{T^{-1}}</math>|| Hz
|-
|<math>F</math>||Drive force||<math>\mathsf{MLT^{-2}}</math>|| N
|-
|<math>g</math>||Acceleration of gravity at the Earth's surface||<math>\mathsf{LT^{-2}}</math>|| m/s<sup>2</sup> 
|-
|<math>i</math>||Imaginary unit, <math>i^2 = -1</math>|| — || — 
|-
|<math>k</math>||Spring constant||<math>\mathsf{MT^{-2}}</math>|| N/m
|-
|<math>\mu</math>||Torsion Spring constant||<math>\mathsf{ML^2T^{-2}}</math>|| Nm/rad
|-
|<math>m, M</math>||Mass||<math>\mathsf{M}</math>|| kg
|-
|<math>Q</math>||Quality factor||—||—
|-
|<math>T</math>||Period of oscillation||<math>\mathsf{T}</math>|| s
|-
|<math>t</math>||Time||<math>\mathsf{T}</math>|| s
|-
|<math>U</math>||Potential energy stored in oscillator||<math>\mathsf{ML^2T^{-2}}</math>|| J
|-
|<math>x</math>||Position of mass||<math>\mathsf{L}</math>|| m
|-
|<math>\zeta</math>||Damping ratio||—|| — 
|-
|<math>\varphi</math>||Phase shift|| — || rad
|-
|<math>\omega</math>||Angular frequency||<math>\mathsf{T^{-1}}</math>|| rad/s
|-
|<math>\omega_0</math>||Natural resonant angular frequency ||<math>\mathsf{T^{-1}}</math>|| rad/s
|}

==See also==
{{cols|colwidth=26em}}
*[[Anharmonicity]]
*[[Critical speed]]
*[[Effective mass (spring–mass system)]]
*[[Fradkin tensor]]
*[[Normal mode]]
*[[Parametric oscillator]]
*[[Phasor]]
*[[Q factor]]
*[[Quantum harmonic oscillator]]
*[[Bertrand's theorem#Harmonic_oscillator|Radial harmonic oscillator]]
*[[Elastic pendulum]]
{{colend}}

==Notes==
{{Reflist}}

==References==
* {{ citation | first1 = Grant R. | last1 = Fowles | first2 = George L. | last2 = Cassiday | year = 1986 | isbn = 0-03-089725-4 | lccn = 93085193 | title = Analytic Mechanics | edition = 5th | publisher = [[Saunders College Publishing]] | location = Fort Worth }}
* {{cite encyclopedia|last1=Hayek|first1=Sabih I.|title=Mechanical Vibration and Damping|encyclopedia=Encyclopedia of Applied Physics|date=15 Apr 2003|doi=10.1002/3527600434.eap231|publisher=WILEY-VCH Verlag GmbH & Co KGaA|isbn=9783527600434}}
* {{citation | last1 = Kreyszig | first1 = Erwin | author-link = Erwin Kreyszig | title = Advanced Engineering Mathematics | edition = 3rd | location = New York | publisher = [[John Wiley & Sons|Wiley]] | year = 1972 | isbn = 0-471-50728-8 | url = https://archive.org/details/advancedengineer00krey }}
* {{cite book | last1 = Serway | first1 = Raymond A. | last2 = Jewett | first2 = John W. | title = Physics for Scientists and Engineers | publisher = Brooks/Cole | year = 2003 | isbn = 0-534-40842-7 | url = https://archive.org/details/physicssciengv2p00serw }}
* {{cite book | last1 = Tipler | first1 = Paul | title = Physics for Scientists and Engineers: Vol. 1 | edition = 4th | publisher = W. H. Freeman | year = 1998 | isbn = 1-57259-492-6 }}
* {{cite book | last1 = Wylie | first1 = C. R. | title = Advanced Engineering Mathematics | url = https://archive.org/details/advancedengineer0000wyli_h4m3 | url-access = registration | edition = 4th | publisher = McGraw-Hill | year = 1975 | isbn = 0-07-072180-7 }}

==External links==
{{Commons category|Harmonic oscillators}}
{{wikiquote}}
*[https://feynmanlectures.caltech.edu/I_21.html The Harmonic Oscillator] from [[The Feynman Lectures on Physics]]


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