Coupled Oscillators and Normal Modes
Two masses joined by springs no longer have one frequency: the motion splits into normal modes, each a pattern of the whole system oscillating at a single frequency. Any starting motion is a combination of those modes.
What a learner can do afterwards
- Writes the coupled equations for two masses and springs and finds both normal mode frequencies
- Describes the in-phase and out-of-phase patterns and which frequency belongs to each
- Expresses a general starting condition as a sum of normal modes
1 · Read
You tuned a single swing to its own frequency. Join two masses with springs and their equations mention each other, so neither mass solves alone. Add and subtract the equations and two special combinations pop out. Each combination oscillates at its own single frequency. Those are the normal modes, and every other motion is just a sum of the two with weights set by the starting push. Frequency belongs to the whole pattern, not one mass.
Name the two patterns. In phase: both masses glide the same way together, the coupling spring never stretches, and this mode runs slower. Out of phase: they move opposite, the coupling spring stretches extra, and this mode runs faster. Each mode alone is ordinary simple harmonic motion, where acceleration points home with strength proportional to displacement.
Start the system in a mixture and the two frequencies beat against each other, so energy sloshes back and forth between the masses. Pull only one mass aside and you excite both modes at once, which makes the motion breathe. Displace both masses equally in the same direction and you get pure in phase motion at the slower frequency.
Reason mode by mode: master the single oscillator and each mode feels familiar. A mode is a pattern of the whole system, never one mass with its own private frequency. Name the pattern first, then attach its frequency.
Coupled motion splits into two whole-system patterns, each ringing at its own frequency.
The rewritten readings were written for this dataset.
More to read
8.4: Coupled Oscillators
Two masses joined by springs move together, so their equations mention each other and cannot be solved separately. Adding and subtracting the equations reveals two special combinations, one for the masses moving as one and one for them moving opposite. Each combination oscillates at its own single frequency, and these are the normal modes. Every other motion is just a sum of the two modes with weights set by the starting push.
Treating each mode as one oscillator
Use suitable combinations of positions to describe each normal mode as a single harmonic oscillator. Its acceleration opposes its displacement and grows in proportion to it. For two identical oscillators joined by a spring, equal motion in phase leaves the coupling spring unchanged. Opposite motion changes its length, giving the out-of-phase mode a higher frequency. Knowing one oscillator helps you analyse both modes.
Energy changes in coupled motion
A single normal mode alternates between kinetic energy and spring potential energy. Their values cycle at twice the mode's frequency. In two identical coupled oscillators, a mixture of modes can produce beats, transferring energy between the masses. Following these energy changes with the equations helps distinguish motion in phase from motion out of phase.
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8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.