Harmonic Oscillator

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 ''damped''. 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 ''driven''.

Mechanical examples include Pendula (with small angles of displacement), masses connected to Spring s, and Acoustical System s. Other analogous systems include electrical harmonic oscillators (see RLC Circuit ).

This article discusses the harmonic oscillator in terms of Classical Mechanics . See the article Quantum Harmonic Oscillator for a discussion of the harmonic oscillator in Quantum Mechanics .

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{d^2x}{dt^2} = -k x

If we define

{\omega_0}^2 = k/m

, then the equation can be written as follows,

:

rac{d^2x}{dt^2} + {\omega_0}^2 x = 0

and has the general solution

:

x = A \cos {(\omega_0 t + \phi)} \,

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 {(\omega_0 t + \phi)} \,

where the value of

\phi \,

is shifted by

\pi/2 \,

relative to the previous form;

or as

:

x = A \sin{\omega_0 t} + B \cos{\omega_0 t} \,

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{\omega_0}{2\pi}

The Kinetic Energy is

:

T = rac{1}{2} m \left(rac{dx}{dt}
ight)^2 = rac{1}{2} k A^2 \sin^2(\omega_0 t + \phi)

.

and the Potential Energy is

:

U = rac{1}{2} k x^2 = rac{1}{2} k A^2 \cos^2(\omega_0 t + \phi)

so the total energy of the system has the constant value

:

E = rac{1}{2} k A^2

DRIVEN HARMONIC OSCILLATOR

This satisfies the nonhomogeneous second order linear differential equation

::

rac{d^2x}{dt^2} + {\omega_0}^2x = A_0 \cos(\omega t).

Example: AC LC (inductor-capacitor) circuit.

Note that this is true if the driving force applied is itself sinusoidal. The term on the right side of the equal sign corresponds to this applied driving force.

DAMPED HARMONIC OSCILLATOR

This satisfies the equation

:

rac{d^2x}{dt^2} + b/m rac{dx}{dt} + {\omega_0}^2x = 0.

Example: weighted spring underwater (where the damping force exerted by the water is proportional to ''b'').

DAMPED, DRIVEN HARMONIC OSCILLATOR

This satisfies the equation

:

mrac{d^2x}{dt^2} + r rac{dx}{dt} + 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 unhomogenous ODE) that is independent of initial conditions and depends only on driving frequency, driving force, restoring force, damping force, and inertial moment of the oscillator (see also Kernel And Image ).

The steady-state solution is

::::

x(t) = rac{F_0}{Z_m \omega} \sin(\omega t - \phi)

where

:

Z_m = \sqrt{r^2 + \left(\omega m - rac{k}{\omega}
ight)^2}

is the absolute value of the Impedance

:

Z = r + i\left(\omega m - rac{k}{\omega}
ight)

and

:

\phi = \arctan\left(rac{\omega m - rac{k}{\omega}}{r}
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

:

{\omega}_r = \sqrt{rac{k}{m} - rac{r^2}{4 m^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:

:

rac{d^2x}{dt^2} + b/m rac{dx}{dt} + {\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(ω''t'') 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{\omega}{2 \pi}.

Important terms

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

= 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 Condition s: 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{d^2x}{dt^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:

::

-k x = ma \,

:where ''k'' is the Spring Constant

m

x

a

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

::

-k x = m rac{d^2 x}{d t^2}.

The most simple solution to the above Differential Equation is

::

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

and the second derivative of that is

::

rac{d^2 x}{dt^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:

::

-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{rac{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
:

rac{d^2q}{d 	au^2} + 2 \zeta rac{dq}{d	au} + 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(ω''τ''), where ω = ω''t_c'', the equation becomes
:

rac{d^2q}{d 	au^2} + 2 \zeta rac{dq}{d	au} + 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 ''c''₁ and ''c''₂ is

& \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:
:

rac{d^2 q}{d	au^2} + 2 \zeta rac{dq}{d	au} + q = \cos(\omega 	au) + i\sin(\omega 	au) = e^{ i \omega 	au} .

Supposing the solution is of the form

:

\,\! q_s(	au) = A e^{i ( \omega 	au + \phi ) } .

Its derivatives from zero to 2nd order are
:

q_s = A e^{i ( \omega 	au + \phi ) }, \ rac{dq_s}{d 	au} = i \omega A e^{i ( \omega 	au + \phi ) }, \ rac{d^2 q_s}{d 	au^2} = - \omega^2 A e^{i ( \omega 	au + \phi ) } .

Substituting these quantities into the differential equation gives

:

\,\! -\omega^2 A e^{i (\omega 	au + \phi)} + 2 \zeta i \omega A e^{i(\omega 	au + \phi)} + A e^{i(\omega 	au + \phi)} = (-\omega^2 A \, + \, 2 \zeta i \omega A \, + \, A) e^{i (\omega 	au + \phi)} = e^{i \omega 	au} .

Dividing by the exponential term on the left results in

:

\,\! -\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
:

A (1-\omega^2)=\cos\phi \qquad 2 \zeta \omega A = - \sin\phi.

Amplitude part

Squaring both equations and adding them together gives
:

\left . \begin{matrix}A^2  (1-\omega^2)^2 = \cos^2\phi \ (2 \zeta \omega A)^2 = \sin^2\phi \end{matrix} 
ight \} \Rightarrow A^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{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
:

an\phi = - rac{2 \zeta \omega}{ 1 - \omega^2} = rac{2 \zeta \omega}{\omega^2 - 1} \Rightarrow \phi \equiv \phi(\zeta, \omega) = \arctan \left( rac{2 \zeta \omega}{\omega^2 - 1} 
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)) = A\cos(\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 RLC circuit, the following correspond:

''F'' ( Force ) $\Leftrightarrow$ ''V'' ( Electric Potential )

''x'' (position) $\Leftrightarrow$ ''Q'' ( Charge )

''k'' ( Spring Constant ) $\Leftrightarrow rac{1}{C}$ ( Electrical Elastance (reciprocal of Capacitance )

''v'' ( Velocity ) $\Leftrightarrow$ ''I'' ( Electric Current )

''b'' ( Damping factor) $\Leftrightarrow$ ''R'' ( Electrical Resistance )

''a'' ( Acceleration ) $\Leftrightarrow rac{dI}{dt} \,$ (rate of change of current)

''m'' ( Mass ) $\Leftrightarrow$ ''L'' ( Inductance )

APPLICATIONS

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

:

V(x) = rac{1}{2} 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{1}{2} (x-x_0)^2 V^{(2)}(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{1}{2} (x-x_0)^2 V^{(2)}(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{1}{2} x^2 V^{(2)}(0) = rac{1}{2} 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.

EXAMPLES

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 given by

:

{d^2	heta\over dt^2}+{g\over \ell}	heta=0

Solution to this equation is given by:

:<math>T 0

2\pi\sqrt{\ell\over g}\quad\quad\quad\quad heta_0 \ll 1</math>

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

x \left( t 
ight) =A\cos \left( (\sqrt {k/m}) 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{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.

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	CATEGORIES ABOUT HARMONIC OSCILLATOR

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Information About

Harmonic Oscillator