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Mechanics · normal modes

Coupled Oscillators

True 3D spring coils connect the masses, velocity arrows show where each bead is heading, and floating sine curves expose the normal modes hidden in the motion. Excite one mode and it stays pure; mix two and beats appear.

Interactive modelCoupled Oscillators
Masses6
Excited mode pp1
Frequency0.000 rad/s
Total energy0.000

Physics tutorial

Coupled oscillators: find normal modes inside complex motion

BackgroundSprings connect neighboring displacements. Certain collective shapes retain their form and oscillate at one frequency.

Why it mattersMolecular vibration, phonons, structural motion, and quantum fields all use the same move: replace coupled coordinates with independent modes.

Start with the essentials

Focus question
Why can N interacting masses still be decomposed into N independent pure motions?
One-sentence intuition
In the right collective coordinates, a coupled system becomes a set of independent harmonic oscillators.

Core mathematical model

Discrete spring chain

mx¨j=k(xj+1−2xj+xj−1)m\ddot{x}_j=k(x_{j+1}-2x_j+x_{j-1})

Mass jj feels the sum of restoring forces from its two neighbors.

Fixed-end normal frequencies

ωp=2kmsin⁡ ⁣(pπ2(N+1))\omega_p=2\sqrt{\frac{k}{m}}\sin\!\left(\frac{p\pi}{2(N+1)}\right)

Each mode pp has its own frequency and fixed node pattern.

Common difficulties

A mode is not one mass

Typical misconceptionMode 2 means the motion of the second mass.

Better mental modelOne mode is a displacement pattern involving every mass.

Beats are not a new eigenfrequency

Typical misconceptionA slow amplitude pulse means a new low-frequency mode appeared.

Better mental modelThe beat is an envelope created by adding two nearby frequencies.

Run the experiment

  1. 01

    Identify the lowest mode

    Choose Mode 1 and show the envelope.

    What to observe: All masses move broadly in the same direction.
  2. 02

    Count nodes

    Compare Mode 2 with Mode 3.

    What to observe: Higher mode number adds internal nodes and raises frequency.
  3. 03

    Superpose two modes

    Choose Beats.

    What to observe: A slow amplitude envelope surrounds fast oscillation.