Harmonic oscillator

Introduction

A harmonic oscillator is a special type of physical system that varies above and below its mean value with a characteristic frequency, f. The system must satisfy that the returning force is proportionate to the displacement, i.e.

$F_{return}=-k\cdot x$

This demand is equivelant to the demand that the system Lagrangian is in the form of

${\mathcal {L}}={\frac {1}{2}}m\left({\frac {dx(t)}{dt}}\right)^{2}-{\frac {1}{2}}k\left(x(t)\right)^{2}$

Note than not every system which oscillates periodically is a harmonic oscliator. One important property of a harmonic oscillator is that the period of the oscillation is independent of the amplitude.

Common examples of harmonic oscillators include pendulums, masses on springs, and RLC circuits.

The following 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.

Full Mathematical Definition

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

${\frac {d^{2}x}{dt^{2}}}+b{\frac {dx}{dt}}+{\omega _{0}}^{2}x=A_{0}\cos(\omega t)$

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:

$f={\frac {\omega }{2\pi }}$

Important terms

• Amplitude: maximal displacement from the equilibrium.
• Period: the time it takes the system to complete an oscillation cycle.
• Frequency: the number of cycles the system performs per unit time (usually measured in Hertz = 1/sec).
• Angular Frequency: $\omega =2\pi \cdot f$
• Phase: how much of a cycle the system complete (system that begins is in phase zero, system which complete half a cycle is in phase $\pi$.
• Initial conditions: the state of the system in t=0, the begining 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:

${\frac {d^{2}x}{dt^{2}}}+{\omega _{0}}^{2}x=0$

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 hanging on a spring, Newton's Laws, combined with Hooke's law for the behavior of a spring, states that:

$-ky=ma$

where k is the spring constant, m is the mass, y is the position of the mass, and a is its acceleration. Rewriting the equation, we obtain:

${\frac {d^{2}y}{dt^{2}}}=-{\frac {k}{m}}y$

The easiest way to solve the above equation is to recognize that when d²z/dt² ∝ -z, z is some form of sine. So we try the solution:

$y=A\cos(\omega t+\delta )$

${\frac {d^{2}y}{dt^{2}}}=-A\omega ^{2}\cos(\omega t+\delta )$

where A is the amplitude, δ is the phase shift, and ω is the angular frequency. Substituting, we have:

$-A\omega ^{2}\cos(\omega t+\delta )=-{\frac {k}{m}}A\cos(\omega t+\delta )$

and thus (dividing both sides by -A cos(ωt + δ)):

$\omega ={\sqrt {\frac {k}{m}}}$

The above formula reveals that the angular frequency 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). That means that what was labelled ω is in fact ω_o. This will become important later.

Driven Harmonic Oscillator

Satisfies equation:

${\frac {d^{2}x}{dt^{2}}}+{\omega _{0}}^{2}x=A_{0}\cos(\omega t)$

Good example:

AC LC circuit.

a few notes about what the response of the circuit to different AC frequencies.

Damped Harmonic Oscillator

Satisfies equation:

${\frac {d^{2}x}{dt^{2}}}+b{\frac {dx}{dt}}+{\omega _{0}}^{2}x=0$

good example:

weighted spring underwater

Note well: underdamped, critically damped

Damped, Driven Harmonic Oscillator

equation:

$m{\frac {d^{2}x}{dt^{2}}}+r{\frac {dx}{dt}}+kx=F_{0}\cos(\omega t)$

The general solution is a sum of a transient (the solution for damped undriven harmonic oscillator, homogenous ODE) and the steady state (particular solution of the unhomogenous ODE). The steady state solution is

$x(t)={\frac {F_{0}}{Z_{m}\cdot \ \omega }}\sin(\omega t-\phi )$

where

$Z_{m}={\sqrt {r^{2}+(\omega \cdot m-{\frac {k}{\omega }})^{2}}}$

is the absolute value of the impedance

$Z=r+i(\omega \cdot m-{\frac {k}{\omega }})$

and

$\phi =\arctan({\frac {\omega m-{\frac {k}{\omega }}}{r}})$

is the phase of the oscillation.

One might see that for certain frequency $\omega$ the amplitude (relative to a given $F_{0}$) is maximal. This occurs for the frequency

${\omega }_{r}={\sqrt {{\frac {k}{m}}-{\frac {r^{2}}{2m^{2}}}}}$

and called Resonance of Displacement.

example:

RLC circuit

Notes for above apply, transient vs steady state response, and quality factor.

A Final Note on Mathematics

For a more complete description of how to solve the above equation, see the article on Differential equations.

See also: normal modes.