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

In classical mechanics, a Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force

F

proportional to the displacement

x

according to Hooke's law:

$F = -k x ,$

where

k

is a positive constant.

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 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 (see RLC circuit).

    Harmonic oscillator
        Simple harmonic oscillator
        Driven harmonic oscillator
        Damped harmonic oscillator
        Damped, driven harmonic oscillator
        Full mathematical definition
            Important terms
        Simple harmonic oscillator
        Universal oscillator equation
            Transient solution
            Steady-state solution
                Amplitude part
                Phase part
            Full solution
        Relationship to RLC circuit
        Applications
            Simple Pendulum
            Pendulum swinging over turntable
            Spring-mass system
        See also

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

$F = -k x ,$

Using Newton's Second Law

$F = m a = -k x ,$

The acceleration,

a

is equal to the second derivative of

x

.

$m rac = -k x$

If we define

^2 = k/m

, then the equation can be written as follows,

$rac + ^2 x = 0$

and has the general solution

$x = A cos ,$

where the amplitude

A ,

and the phase

phi ,

are determined by the initial conditions.

Alternatively, the general solution can be written as

$x = A sin ,$

where the value of

phi ,

is shifted by

pi/2 ,

relative to the previous form;

or as

$x = A sin + B cos ,$

where

A ,

and

B ,

are the constants which are determined by the initial conditions, instead of

A ,

and

phi ,

in the previous forms.

The frequency of the oscillations is given by

$f = rac$

The kinetic energy is

$T = rac m left(rac$

ight)^2 = rac k A^2 sin^2(omega_0 t + phi).

and the potential energy is

$U = rac k x^2 = rac k A^2 cos^2(omega_0 t + phi)$

so the total energy of the system has the constant value

$E = rac k A^2$

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

$rac + ^2x = A_0 cos(omega t),$

where

A_

is the driving amplitude and

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
A damped harmonic oscillator satisfies the second order differential equation

$rac + rac rac + ^2x = 0,$

where

b

is an experimentally determined damping constant satisfying the relationship

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

b

.

Damped, driven harmonic oscillator
This satisfies the equation

$mrac + r rac + kx= F_0 cos(omega t).$

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

$x(t) = rac sin(omega t - phi)$

where

$Z_m = sqrt$

is the absolute value of the impedance

$Z = r + ileft(omega m - rac$

ight)

and

$phi = arctanleft(rac$

ight)

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

One might see that for a certain driving frequency,

omega

, the amplitude (relative to a given

F_0

) is maximal. This occurs for the frequency

$_r = sqrt$

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:

$rac + b/m rac + ^2x = A_0 cos(omega t)$

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

$f = rac.$

Important terms

Amplitude: maximal displacement from the equilibrium.

Period: the time it takes the system to complete an oscillation cycle. Opposite of frequency.

Frequency: the number of cycles the system performs per unit time (usually measured in hertz = 1/s).

Angular frequency:

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

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:

$rac + ^2x = 0.$

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:

$-k x = ma ,$

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:

$-k x = m rac.$

The most simple solution to the above differential equation is

$x = A cos(omega t + delta) ,$

and the second derivative of that is

$rac = -A omega^2 cos(omega t + delta)$

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:

$-A k cos(omega t +delta) = -A m omega^2 cos(omega t + delta). ,$

Then, after dividing both sides by

-A cos(omega t + delta) ,

we get:

$k = m omega^2 ,$

or, as it is more commonly written:

$omega = sqrt.$

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

$rac + 2 zeta rac + q = 0$

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

$rac + 2 zeta rac + q = cos(omega au).$

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

q_t (	au) = egin e^ left( c_1 e^ + c_2 e^ 
ight) & zeta > 1  mbox \ e^ (c_1+c_2 	au) = e^(c_1+c_2 	au) & zeta = 1  mbox \ e^ left c_1 cos left(sqrt{1-zeta^2} 	au
ight) +c_2 sinleft(sqrt{1-zeta^2} 	au
ight) 
ight & zeta < 1  mbox end

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:

$rac + 2 zeta rac + q = cos(omega au) + isin(omega au) = e^ .$

Supposing the solution is of the form

$,! q_s( au) = A e^ .$

Its derivatives from zero to 2nd order are

$q_s = A e^, rac = i omega A e^, rac = - omega^2 A e^ .$

Substituting these quantities into the differential equation gives

$,! -omega^2 A e^ + 2 zeta i omega A e^ + A e^ = (-omega^2 A , + , 2 zeta i omega A , + , A) e^ = e^ .$

Dividing by the exponential term on the left results in

$,! -omega^2 A + 2 zeta i omega A + A = e^ = cosphi - i sinphi .$

Equating the real and imaginary parts results in two independent equations

$A (1-omega^2)=cosphi qquad 2 zeta omega A = - sinphi.$

Amplitude part
Squaring both equations and adding them together gives

$left . eginA^2 (1-omega^2)^2 = cos^2phi \ (2 zeta omega A)^2 = sin^2phi end$

ight } 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,

$A = A( zeta, omega) = rac.$

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

$anphi = - rac = rac Rightarrow phi equiv phi(zeta, omega) = arctan left( rac$

ight ).

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

$,! q_s ( au) = A(zeta,omega) cos(omega au + phi(zeta,omega)) = Acos(omega au + phi).$

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

$,! q( au) = q_t ( au) + q_s ( au).$

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

Relationship to RLC circuit
Comparing a mechanical harmonic oscillator with an series RLC circuit or parallel RLC circuit, the following correspond:

F (force)

Leftrightarrow

V (electric potential)

Leftrightarrow

I (electric current)

x (position)

Leftrightarrow

Q (charge)

Leftrightarrow

Φ (magnetic flux)

k (spring constant)

Leftrightarrow rac

(electrical elastance (reciprocal of capacitance)

Leftrightarrow rac

(reciprocal of inductance)

Leftrightarrow

I (electric current)

Leftrightarrow

V (electric potential)

b (damping factor)

Leftrightarrow

R (electrical resistance)

Leftrightarrow rac

(reciprocal of electrical resistance)

a (acceleration)

Leftrightarrow rac ,

(rate of change of current)

Leftrightarrow rac ,

(rate of change of electric potential)

m (mass)

Leftrightarrow

Leftrightarrow

C (capacitance)

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

$V(x) = rac k x^2.$

Given an arbitrary potential energy function

V(x)

, one can do a Taylor expansion in terms of

x

around an energy minimum (

x = x_0

) to model the behavior of small perturbations from equilibrium.

$V(x) = V(x_0) + (x-x_0) V'(x_0) + rac (x-x_0)^2 V^(x_0) + O(x-x_0)^3$

Because

V(x_0)

is a minimum, the first derivative evaluated at

x_0

must be zero, so the linear term drops out:

$V(x) = V(x_0) + rac (x-x_0)^2 V^(x_0) + O(x-x_0)^3$

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

$V(x) approx rac x^2 V^(0) = rac k x^2$

Thus, given an arbitrary potential energy function

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.

Simple Pendulum

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

$+ heta=0$

Solution to this equation is given by:

$heta(t) = heta_0cosleft(sqrtt$

ight) quadquadquadquad | heta_0| ll 1

where

heta_0

is the largest angle attained by the pendulum. Period, the time for one complete oscillation (time for the bob to return to its starting position), is given by

2pi

divided by whatever is multiplying the time in the argument of the cosine

$T_0 = 2pisqrtquadquadquadquad | heta_0| ll 1$

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

omega=2picdot f = rac

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. The Hooke's Law gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length.

$Fs left( t
ight) =kx left( t
ight)$

where Fs is the force, k is the spring constant, and the 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:

$m rac x left( t
ight) +kx(t)=0$

If the initial displacement is A, and there is no initial velocity, the solution of this equation is given by:
$x left( t
ight) =Acos left( (sqrt ) t
ight)$

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

U = 1/2,k^

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.

See also

Quantum harmonic oscillator

Anharmonic oscillator

Parametric oscillator

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