Harmonic oscillator

This article is about the harmonic oscillator in classical mechanics. For its use in quantum mechanics, see quantum harmonic oscillator.

An undamped spring-mass system is a simple harmonic oscillator.

In classical mechanics, a harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force Failed to parse (Missing texvc executable; please see math/README to configure.): F

proportional to the displacement Failed to parse (Missing texvc executable; please see math/README to configure.): x
according to Hooke's law:

Failed to parse (Missing texvc executable; please see math/README to configure.): F = -k x \,

where Failed to parse (Missing texvc executable; please see math/README to configure.): k

is a positive constant.

If Failed to parse (Missing texvc executable; please see math/README to configure.): F

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

If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. In such situation, the frequency of the oscillations is smaller than in the non-damped case, and the amplitude of the oscillations decreases with time.

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

Mechanical examples include pendula (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits (see Equivalent systems below).

Simple harmonic oscillator

The simple harmonic oscillator has no driving force, and no friction (damping), so the net force is just:

Failed to parse (Missing texvc executable; please see math/README to configure.): F = -k x \

Using Newton's Second Law of motion,

Failed to parse (Missing texvc executable; please see math/README to configure.): F = m a = -k x \,

The acceleration, Failed to parse (Missing texvc executable; please see math/README to configure.): a

is equal to the second derivative of Failed to parse (Missing texvc executable; please see math/README to configure.): x

.

Failed to parse (Missing texvc executable; please see math/README to configure.): m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} = -k x

If we define Failed to parse (Missing texvc executable; please see math/README to configure.): {\omega_0}^2 = k/m , then the equation can be written as follows,

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + {\omega_0}^2 x = 0

Define Failed to parse (Missing texvc executable; please see math/README to configure.): \dot x = \frac{\mathrm{d}x}{\mathrm{d}t} .
We observe that:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d}^2 x}{\mathrm{d} t^2} = \ddot x = \frac{\mathrm{d}\dot {x}}{\mathrm{d}t}\frac{\mathrm{d}x}{\mathrm{d}x}=\frac{\mathrm{d}\dot {x}}{\mathrm{d}x}\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{\mathrm{d}\dot{x}}{\mathrm{d}x}\dot {x}

and substituting

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d} \dot{x}}{\mathrm{d}x}\dot x + {\omega_0}^2 x = 0

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d} \dot{x}\cdot \dot x + {\omega_0}^2 x \cdot \mathrm{d}x = 0

integrating

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{x}^2 + {\omega_0}^2 x^2 = K

where K is the integration constant, set K = (A ω₀)²

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{x}^2 = A^2 {\omega_0}^2-{\omega_0}^2 x^2

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{x} = \pm {\omega_0} \sqrt{A^2 - x^2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac {\mathrm{d}x}{\pm \sqrt{A^2 - x^2}} = {\omega_0}\mathrm{d}t

integrating, the results (including integration constant ϕ) are

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{cases} \arcsin{\frac {x}{A}}= \omega_0 t + \phi \\ \arccos{\frac {x}{A}}= \omega_0 t + \phi \end{cases}

and has the general solution

Failed to parse (Missing texvc executable; please see math/README to configure.): x = A \cos {(\omega_0 t + \phi)} \,

where the amplitude Failed to parse (Missing texvc executable; please see math/README to configure.): A \,

and the phase Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \,
are determined by the initial conditions.

Alternatively, the general solution can be written as

Failed to parse (Missing texvc executable; please see math/README to configure.): x = A \sin {(\omega_0 t + \phi)} \,

where the value of Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \,

is shifted by Failed to parse (Missing texvc executable; please see math/README to configure.): \pi/2 \,
relative to the previous form;

or as

Failed to parse (Missing texvc executable; please see math/README to configure.): x = A \sin{\omega_0 t} + B \cos{\omega_0 t} \,

where Failed to parse (Missing texvc executable; please see math/README to configure.): A \,

and Failed to parse (Missing texvc executable; please see math/README to configure.): B \,
are the constants which are determined by the initial conditions, instead of Failed to parse (Missing texvc executable; please see math/README to configure.): A \,
and Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \,
in the previous forms.

The frequency of the oscillations is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle f = \frac {\omega_0} {2 \pi} = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}

The kinetic energy is

Failed to parse (Missing texvc executable; please see math/README to configure.): T = \frac{1}{2} m \left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2 = \frac{1}{2} k A^2 \sin^2(\omega_0 t + \phi)

.

and the potential energy is

Failed to parse (Missing texvc executable; please see math/README to configure.): U = \frac{1}{2} k x^2 = \frac{1}{2} k A^2 \cos^2(\omega_0 t + \phi)

so the total energy of the system has the constant value

Failed to parse (Missing texvc executable; please see math/README to configure.): E = \frac{1}{2} k A^2.

Driven harmonic oscillator

A driven harmonic oscillator satisfies the nonhomogeneous second order linear differential equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + {\omega_0}^2x = A_0 \cos(\omega t),

where Failed to parse (Missing texvc executable; please see math/README to configure.): A_{0}

is the driving amplitude and Failed to parse (Missing texvc executable; please see math/README to configure.): \omega
is the driving frequency for a sinusoidal driving mechanism. This type of system appears in AC LC (inductor-capacitor) circuits and idealized spring systems lacking internal mechanical resistance or external air resistance.

Damped harmonic oscillator

Main article: Damping

A damped harmonic oscillator satisfies the second order differential equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + \frac{b}{m} \frac{\mathrm{d}x}{\mathrm{d}t} + {\omega_0}^2x = 0,

where Failed to parse (Missing texvc executable; please see math/README to configure.): b

is an experimentally determined damping constant satisfying the relationship Failed to parse (Missing texvc executable; please see math/README to configure.): F = -bv

. An example of a system obeying this equation would be a weighted spring underwater if the damping force exerted by the water is assumed to be linearly proportional to Failed to parse (Missing texvc executable; please see math/README to configure.): v .

The frequency of the damped harmonic oscillator is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \omega_1 = \sqrt{\omega_0^2 - R_m^2}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): R_m=\frac{b}{2m}.

Damped, driven harmonic oscillator

This satisfies the equation

Failed to parse (Missing texvc executable; please see math/README to configure.): m\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + r \frac{\mathrm{d}x}{\mathrm{d}t} + kx= F_0 \cos(\omega t).

The general solution is a sum of a transient (the solution for damped undriven harmonic oscillator, homogeneous ODE) that depends on initial conditions, and a steady state (particular solution of the nonhomogenous ODE) that is independent of initial conditions and depends only on driving frequency, driving force, restoring force, damping force,

The steady-state solution is

Failed to parse (Missing texvc executable; please see math/README to configure.): x(t) = \frac{F_0}{Z_m \omega} \sin(\omega t - \phi)

where

Failed to parse (Missing texvc executable; please see math/README to configure.): Z_m = \sqrt{r^2 + \left(\omega m - \frac{k}{\omega}\right)^2}

is the absolute value of the impedance or linear response function

Failed to parse (Missing texvc executable; please see math/README to configure.): Z = r + i\left(\omega m - \frac{k}{\omega}\right)

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \phi = \arctan\left(\frac{\omega m - \frac{k}{\omega}}{r}\right)

is the phase of the oscillation relative to the driving force.

One might see that for a certain driving frequency, Failed to parse (Missing texvc executable; please see math/README to configure.): \omega , the amplitude (relative to a given Failed to parse (Missing texvc executable; please see math/README to configure.): F_0 ) is maximal. This occurs for the frequency

Failed to parse (Missing texvc executable; please see math/README to configure.): {\omega}_r = \sqrt{\frac{k}{m} - 2\left(\frac{r}{2 m}\right)^2}

and is called resonance of displacement.

In summary: at a steady state the frequency of the oscillation is the same as that of the driving force, but the oscillation is phase-offset and scaled by amounts that depend on the frequency of the driving force in relation to the preferred (resonant) frequency of the oscillating system.

Example: RLC circuit.

Full mathematical definition

Most harmonic oscillators, at least approximately, solve the differential equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + \frac{b}{m} \frac{\mathrm{d}x}{\mathrm{d}t} + {\omega_0}^2x = A_0 \cos(\omega t)

where t is time, b is the damping constant, ω_o is the characteristic angular frequency, and A_ocos(ωt) represents something driving the system with amplitude A_o and angular frequency ω. x is the measurement that is oscillating; it can be position, current, or nearly anything else. The angular frequency is related to the frequency, f, by

Failed to parse (Missing texvc executable; please see math/README to configure.): f = \frac{\omega}{2 \pi}.

Important terms

• Amplitude: maximal displacement from the equilibrium.
• Period: the time it takes the system to complete an oscillation cycle. Inverse of frequency.
• Frequency: the number of cycles the system performs per unit time (usually measured in hertz = 1/s).
• Angular frequency: Failed to parse (Missing texvc executable; please see math/README to configure.): \omega = 2 \pi f

• Phase: how much of a cycle the system completed (system that begins is in phase zero, system which completed half a cycle is in phase Failed to parse (Missing texvc executable; please see math/README to configure.): \pi

).

• Initial conditions: the state of the system at t = 0, the beginning of oscillations.

Simple harmonic oscillator

A simple harmonic oscillator is simply an oscillator that is neither damped nor driven. So the equation to describe one is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + {\omega_0}^2x = 0.

Physically, the above never actually exists, since there will always be friction or some other resistance, but two approximate examples are a mass on a spring and an LC circuit.

In the case of a mass attached to a spring, Newton's Laws, combined with Hooke's law for the behavior of a spring, states that:

Failed to parse (Missing texvc executable; please see math/README to configure.): -k x = ma \,

where k is the spring constant

m is the mass

x is the position of the mass

a is its acceleration.

Because acceleration a is the second derivative of position x, we can rewrite the equation as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): -k x = m \frac{\mathrm{d}^2 x}{\mathrm{d} t^2}.

The most simple solution to the above differential equation is

Failed to parse (Missing texvc executable; please see math/README to configure.): x = A \cos(\omega t + \delta) \,

and the second derivative of that is

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -A \omega^2 \cos(\omega t + \delta)

where A is the amplitude, δ is the phase shift, and ω is the angular frequency.

Plugging these back into the original differential equation, we have:

Failed to parse (Missing texvc executable; please see math/README to configure.): -A k \cos(\omega t +\delta) = -A m \omega^2 \cos(\omega t + \delta). \,

Then, after dividing both sides by Failed to parse (Missing texvc executable; please see math/README to configure.): -A \cos(\omega t + \delta) \,

we get:

Failed to parse (Missing texvc executable; please see math/README to configure.): k = m \omega^2 \,

or, as it is more commonly written:

Failed to parse (Missing texvc executable; please see math/README to configure.): \omega = \sqrt{\frac{k}{m}}.

The above formula reveals that the angular frequency ω of the solution is only dependent upon the physical characteristics of the system, and not the initial conditions (those are represented by A and δ). We will label this ω as ω_o from now on. This will become important later.

Universal oscillator equation

The equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d}^2q}{\mathrm{d} \tau^2} + 2 \zeta \frac{\mathrm{d}q}{\mathrm{d}\tau} + q = 0

is known as the universal oscillator equation since all second order linear oscillatory systems can be reduced to this form. This is done through nondimensionalization.

If the forcing function is f(t) = cos(ωt) = cos(ωt_cτ) = cos(ωτ), where ω = ωt_c, the equation becomes

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d}^2q}{\mathrm{d} \tau^2} + 2 \zeta \frac{\mathrm{d}q}{\mathrm{d}\tau} + q = \cos(\omega \tau).

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 c₁ and c₂ is

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 \ \mbox{(overdamping)} \\ e^{-\zeta\tau} (c_1+c_2 \tau) = e^{-\tau}(c_1+c_2 \tau) & \zeta = 1 \ \mbox{(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 \ \mbox{(underdamping)} \end{cases}

The transient solution is independent of the forcing function. If the system is critically damped, the response is independent of the damping.

Steady-state solution

Apply the "complex variables method" by solving the auxiliary equation below and then finding the real part of its solution:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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} .

Supposing the solution is of the form

Failed to parse (Missing texvc executable; please see math/README to configure.): \,\! q_s(\tau) = A e^{i ( \omega \tau + \phi ) } .

Its derivatives from zero to 2nd order are

Failed to parse (Missing texvc executable; please see math/README to configure.): q_s = A e^{i ( \omega \tau + \phi ) }, \ \frac{\mathrm{d}q_s}{\mathrm{d} \tau} = i \omega A e^{i ( \omega \tau + \phi ) }, \ \frac{\mathrm{d}^2 q_s}{\mathrm{d} \tau^2} = - \omega^2 A e^{i ( \omega \tau + \phi ) } .

Substituting these quantities into the differential equation gives

Failed to parse (Missing texvc executable; please see math/README to configure.): \,\! -\omega^2 A e^{i (\omega \tau + \phi)} + 2 \zeta i \omega A e^{i(\omega \tau + \phi)} + A e^{i(\omega \tau + \phi)} = (-\omega^2 A \, + \, 2 \zeta i \omega A \, + \, A) e^{i (\omega \tau + \phi)} = e^{i \omega \tau} .

Dividing by the exponential term on the left results in

Failed to parse (Missing texvc executable; please see math/README to configure.): \,\! -\omega^2 A + 2 \zeta i \omega A + A = e^{-i \phi} = \cos\phi - i \sin\phi .

Equating the real and imaginary parts results in two independent equations

Failed to parse (Missing texvc executable; please see math/README to configure.): A (1-\omega^2)=\cos\phi \qquad 2 \zeta \omega A = - \sin\phi.

Amplitude part

Squaring both equations and adding them together gives

Failed to parse (Missing texvc executable; please see math/README to configure.): \left . \begin{matrix}A^2 (1-\omega^2)^2 = \cos^2\phi \\ (2 \zeta \omega A)^2 = \sin^2\phi \end{matrix} \right \} \Rightarrow A^2[(1-\omega^2)^2 + (2 \zeta \omega)^2] = 1.

By convention the positive root is taken since amplitude is usually considered a positive quantity. Therefore,

Failed to parse (Missing texvc executable; please see math/README to configure.): A = A( \zeta, \omega) = \frac{1}{\sqrt{(1-\omega^2)^2 + (2 \zeta \omega)^2}}.

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.

Note that the variables in these equations ought to be identified before showing the equation.

Phase part

To solve for φ, divide both equations to get

Failed to parse (Missing texvc executable; please see math/README to configure.): \tan\phi = - \frac{2 \zeta \omega}{ 1 - \omega^2} = \frac{2 \zeta \omega}{\omega^2 - 1} \Rightarrow \phi \equiv \phi(\zeta, \omega) = \arctan \left( \frac{2 \zeta \omega}{\omega^2 - 1} \right ).

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \,\! q_s (\tau) = A(\zeta,\omega) \cos(\omega \tau + \phi(\zeta,\omega)) = A\cos(\omega \tau + \phi).

The solution of original universal oscillator equation is a superposition (sum) of the transient and steady-state solutions

Failed to parse (Missing texvc executable; please see math/README to configure.): \,\! q(\tau) = q_t (\tau) + q_s (\tau).

For a more complete description of how to solve the above equation, see linear ODEs with constant coefficients.

Equivalent systems

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 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 will be the same.

Applications

The problem of the simple harmonic oscillator occurs frequently in physics because of the form of its potential energy function:

Failed to parse (Missing texvc executable; please see math/README to configure.): V(x) = \frac{1}{2} k x^2.

Given an arbitrary potential energy function Failed to parse (Missing texvc executable; please see math/README to configure.): V(x) , one can do a Taylor expansion in terms of Failed to parse (Missing texvc executable; please see math/README to configure.): x

around an energy minimum (Failed to parse (Missing texvc executable; please see math/README to configure.): x = x_0

) to model the behavior of small perturbations from equilibrium.

Failed to parse (Missing texvc executable; please see math/README to configure.): V(x) = V(x_0) + (x-x_0) V'(x_0) + \frac{1}{2} (x-x_0)^2 V^{(2)}(x_0) + O(x-x_0)^3

Because Failed to parse (Missing texvc executable; please see math/README to configure.): V(x_0)

is a minimum, the first derivative evaluated at Failed to parse (Missing texvc executable; please see math/README to configure.): x_0
must be zero, so the linear term drops out:

Failed to parse (Missing texvc executable; please see math/README to configure.): V(x) = V(x_0) + \frac{1}{2} (x-x_0)^2 V^{(2)}(x_0) + O(x-x_0)^3

The constant term V(x₀) is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved:

Failed to parse (Missing texvc executable; please see math/README to configure.): V(x) \approx \frac{1}{2} x^2 V^{(2)}(0) = \frac{1}{2} k x^2

Thus, given an arbitrary potential energy function Failed to parse (Missing texvc executable; please see math/README to configure.): V(x)

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

A simple pendulum exhibits simple harmonic motion under the conditions of no damping and small amplitude.

Assuming no damping and small amplitudes, the differential equation governing a simple pendulum is

Failed to parse (Missing texvc executable; please see math/README to configure.): {\mathrm{d}^2\theta\over \mathrm{d}t^2}+{g\over \ell}\theta=0.

The solution to this equation is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \theta(t) = \theta_0\cos\left(\sqrt{g\over \ell}t\right) \quad\quad\quad\quad |\theta_0| \ll 1

where Failed to parse (Missing texvc executable; please see math/README to configure.): \theta_0

is the largest angle attained by the pendulum.  The period, the time for one complete oscillation , is given by Failed to parse (Missing texvc executable; please see math/README to configure.): 2\pi
divided by whatever is multiplying the time in the argument of the cosine

Failed to parse (Missing texvc executable; please see math/README to configure.): T_0 = 2\pi\sqrt{\ell\over g}\quad\quad\quad\quad |\theta_0| \ll 1.

Pendulum swinging over turntable

Simple harmonic motion can in some cases be considered to be the one-dimensional projection of two-dimensional circular motion. Consider a long pendulum swinging over the turntable of a record player. On the edge of the turntable there is an object. If the object is viewed from the same level as the turntable, a projection of the motion of the object seems to be moving backwards and forwards on a straight line. It is possible to change the frequency of rotation of the turntable in order to have a perfect synchronization with the motion of the pendulum.

The angular speed of the turntable is the pulsation of the pendulum.

In general, the pulsation-also known as angular frequency, of a straight-line simple harmonic motion is the angular speed of the corresponding circular motion.

Therefore, a motion with period T and frequency f=1/T has pulsation

Failed to parse (Missing texvc executable; please see math/README to configure.): \omega=2\pi f = \frac{2\pi}{T}.

In general, pulsation and angular speed are not synonymous. For instance the pulsation of a pendulum is not the angular speed of the pendulum itself, but it is the angular speed of the corresponding circular motion.

Spring-mass system

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): F \left( t \right) =kx \left( t \right)

where F is the force, k is the spring constant, and x is the displacement of the mass with respect to the equilibrium position.

This relationship shows that the distance of the spring is always opposite to the force of the spring.

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): m \frac {\mathrm{d}^{2}}{\mathrm{d}{t}^{2}} x \left( t \right) +kx(t)=0.

If the initial displacement is A, and there is no initial velocity, the solution of this equation is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): x \left( t \right) =A\cos \left( (\sqrt {k/m}) t\right).

Energy variation in the spring-damper 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 then is transferred into kinetic energy. The potential energy within a spring is determined by the equation Failed to parse (Missing texvc executable; please see math/README to configure.): U = 1/2\,k{x}^{2}.

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 maximum potential energy, the kinetic energy of the mass is zero. When the spring is released, the spring will try to reach back to equilibrium, and all its potential energy is converted into kinetic energy of the mass.

References

• Serway, Raymond A.; Jewett, John W. (2003). Physics for Scientists and Engineers. Brooks/Cole. ISBN 0-534-40842-7. 
• Tipler, Paul (1998). Physics for Scientists and Engineers: Vol. 1, 4th ed., W. H. Freeman. ISBN 1-57259-492-6. 
• Wylie, C. R. (1975). Advanced Engineering Mathematics, 4th ed., McGraw-Hill. ISBN 0-07-072180-7. 

See also

• Q factor
• Normal mode
• Quantum harmonic oscillator
• Anharmonic oscillator
• Parametric oscillator
• Critical speed

External links

• Harmonic Oscillator from The Chaos Hypertextbook
• Simple Harmonic oscillator on PlanetPhysicsar:هزاز توافقي

da:Harmonisk oscillator de:Harmonischer Oszillator es:Oscilador armónico fr:Oscillateur harmonique it:moto armonico he:אוסצילטור הרמוני hr:Harmonijsko titranje ja:調和振動子 pl:Oscylator harmoniczny pt:Movimento oscilatório ru:Гармонический осциллятор sl:Nihanje sv:Harmonisk oscillator fi:Harmoninen värähtelijä uk:Гармонічний осцилятор

• A-level Physics experiment on the subject of Damped Harmonic Motion with solution curve graphs