Harmonic oscillator

Introduction

A harmonic oscillator is any physical system that varies above and below its mean value with a characteristic frequency, f. Common examples of harmonic oscillators include pendulums, masses on springs, and LRC circuits.

Full Mathematical Definition

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

d²x/dt² - b dx/dt + (ω_o)²x = A_ocos(ωt)

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 = ω/(2π)

Although the above is all there is to it, it's hardly the whole story.

Simple Harmonic Oscillator

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

d²x/dt² + (ω_o)²x = 0

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:

d²y/dt² = -(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(ωt + δ)
d²y/dt² = -Aω²cos(ωt + δ)

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

-Aω²cos(ωt + δ) = -(k/m) A cos(ωt + δ)

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

ω = √(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 labled ω is in fact ω_o. This will become important later.

Driven Harmonic Oscillator

Satisfies equation:

d²x/dt² + (ω_o)²x = A_ocos(ωt)

Good example:

AC LC circuit.

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

Damped Harmonic Oscillator

Satisfies equation:

d²x/dt² - b dx/dt + (ω_o)²x = 0

good example:

weighted spring underwater

Note well: underdamped, critically damped

Damped, Driven Harmonic Oscillator

equation:

d²x/dt² - b dx/dt + (ω_o)²x = A_ocos(ωt)

example:

RLC circuit

Notes for above apply, transient vs stead 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.