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# Harmonic oscillator

## Introduction

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:

where t is time, b is the damping constant, ωo is the characteristic angular frequency, and Aocos(ωt) represents something driving the system with amplitude Ao 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:

### Simple Harmonic Oscillator

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

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:

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:

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:

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

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

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 ωo. This will become important later.

### Driven Harmonic Oscillator

Satisfies equation:

Good example:

AC LC circuit.

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

### Damped Harmonic Oscillator

Satisfies equation:

good example:

weighted spring underwater

Note well: underdamped, critically damped

### Damped, Driven Harmonic Oscillator

equation:

example:

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