Coupled oscillators

Note that the simulation must be re-started for any changes to take effect.

The simulation above shows several identical masses (in red) being driven by a sinusoidally-oscillating mass on the left (in black). The masses are joined together by identical springs; the springs are not drawn so you have to use your imagination a little.

Once you have set the parameters the way you want them, try varying the angular frequency to see if you can find the frequencies of the normal modes of oscillation. For N red masses, there are N normal modes, each with its own frequency.

A normal mode is one of the natural modes of oscillation of the system. It could be all the masses oscillating in phase (all going left at the same time, and then all going right again), or perhaps half the masses going left while the other half go right.

The angular frequencies of the normal modes, with no damping, are given by the equation:

Note that the equation gives the square of the angular frequency; remember to take the square root!

The damping is essentially a frictional effect. Adding damping reduces the normal mode frequencies slightly. Damping is important, though - without it you can't isolate a single normal mode. The starting configuration of the masses, all at their equilibrium position, is a superposition of the different normal modes. You can only eliminate the ones you don't want by taking them away with the damping.

Things to look for:

• With one oscillating mass, compare the amplitude of oscillation at resonance (when the mass is being driven at its natural frequency of oscillation) with the amplitude at a lower or higher frequency. Also, look at the phase relationship between the driver and the oscillating mass above, below, and at resonance.
• With more than one mass, see if you can find all the different normal modes.
• Change the parameters one at a time to see if you can figure out what role each plays.

  Created by Andrew Duffy, Boston University Physics Department.

  Last update March 16, 1998