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# Coupled Oscillators – Two masses, Three Springs

When two or more objects undergoing harmonic motion are connected, we classify them as coupled oscillators.  The resulting motion can become relatively complicated.  However, a few steps can help simplify the the analysis.

## Coordinate System for Two Coupled Oscillators

Coordinate choice for two coupled oscillators.

As usual, when working with an oscillator, the obvious choice of coordinate system is at the equilibrium point.  However, with multiple oscillators, each has its own equilibrium point.  This makes it necessary to measure from multiple locations, something you may not have had to do before.  In the diagram above, we measure the position of each of the masses relative to its equilibrium point.

This allows us to determine the stretch or compression of each spring.  For example, the left hand spring has been stretched by an amount

$$\Delta \ell =\ell_o + x_1$$

The following video goes into a bit more detail.

## Coupled Oscillator Coordinates Video

Once the coordinates are set up, we can use Newton’s Second Law to relate the forces caused by the springs to the accelerations of each of the objects.   When we look at the resulting equations, each mass is dependent on both of the coordinates.

Applying a bit of linear algebra, we can determine the normal modes or eigenmodes of the system as a whole, as well as the eigenfrequencies.

The following video details this process.