# Harmonic oscillator

A**harmonic oscillator**is a mechanical system in which there exists a returning force

*F*directly proportionate to the displacement

*x*, i.e.

*k*is a constant. It can also refer to any physical system that is analogous to this mechanical system, in which some other quantity behaves in the same way as

*x*. Examples of harmonic oscillators include pendulums (in small angles), masses on springss, and RLC circuits.

If *F* is the only force acting on the mechanical system, the system is called a **simple harmonic oscillator**. The motion of a simple harmonic oscillator, called simple harmonic motion, is essentially a sine function oscillating about the equilibrium displacement, *x = 0*, at which the returning force is zero.

The potential energy *V* associated with such a returning force is called a **harmonic potential**. It has the form

## Full mathematical definition

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

where*t*is time,

*b*is the damping constant, ω

_{o}is the characteristic angular frequency, and

*A*

_{o}cos(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:

### 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/s).
- Angular frequency:
- 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 ).
- 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:

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:

*k*is the spring constant,

*m*is the mass,

*y*is the position of the mass, and

*a*is its acceleration. Noting that acceleration is the second derivative of position, we can rewrite the equation as follows:

*d*

^{2}

*z*/

*dt*

^{2}∝ -

*z*,

*z*is some form of sine. So we try the solution:

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

*A*cos(ω

*t*+ δ)):

*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:

AC LC circuit.

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

### Damped harmonic oscillator

Satisfies equation:

weighted spring underwater

Note well: underdamped, critically damped

### Damped, driven harmonic oscillator

equation:

One might see that for certain frequency the amplitude (relative to a given ) is maximal. This occurs for the frequency

**resonance of displacement**.

example:

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