Definition:Simple Harmonic Motion

This page is about Simple Harmonic Motion. For other uses, see Harmonic.

Definition

Consider a physical system $S$ whose motion can be expressed in the form of the following equation:

$x = A \map \sin {\omega t + \phi}$

where $A$ and $\phi$ are constants.

Then $S$ is in a state of simple harmonic motion.

The parameter $A$ is known as the amplitude of the motion.

The period $T$ of the motion of $S$ is the time required for one complete cycle:

$T = \dfrac {2 \pi} \omega$

The frequency $\nu$ of the motion of $S$ is the number of complete cycles per unit time:

$\nu = \dfrac 1 T = \dfrac \omega {2 \pi}$

Simple harmonic motion can also be characterised in the form:

$x = A \map \cos {\omega t + \phi}$

$\map \sin {\omega t + \phi + \dfrac \pi 2} = \map \cos {\omega t + \phi}$

the two forms can be seen to be equivalent.

Simple harmonic motion can also be referred to as simple harmonic oscillation or simple harmonic vibration.

Some sources abbreviate it to SHM.

1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.20$: Vibrations in Mechanical Systems