M065 · Normal modes

Coupled Oscillators

Can one bead's swing grow and fade while the energy in every normal mode stays fixed? It can. Written in normal modes, the chain splits into independent oscillators, and each one keeps its own energy forever. A single bead, though, moves with a mix of modes. When two of those modes have nearby frequencies, they drift in and out of step, and the bead's swing grows and fades: a beat.

Release a chain of beads and springs with fixed ends, either in one sine-shaped mode or in two neighboring high modes; the beads move along the chain. Pick any bead and compare the forces from its two springs with its acceleration, follow how each mode adds to its displacement, and track the energy computed directly from bead positions and velocities. The three-dimensional apparatus, the event cursor and the plots all show the same exact solution.

Exact normalized cursor $0$

Masses $0$

Excited modes $0$

First normalized angular frequency $0$

Second normalized angular frequency $0$

Coordinate-space energy $0$

Independent modal energy $0$

Signed coordinate minus modal energy $0$

Absolute beat-envelope period $0$

Inspected bead $0$

Inspected displacement $0$

Inspected velocity $0$

Force from left spring $0$

Force from right spring $0$

Inspected acceleration $0$

Track one bead and the energy in each mode

Separate local motion from modal energy

Spring coilsSchematic coils; physical force uses model extension.Prepared releaseGold outlines mark the initial positions.Two spring forces and resultantAll force arrows use one fixed scale.Inspected velocityBlue arrow, drawn on its own fixed scale.Mass count

N

6.00\,

Single mode number

p

1.00\,

Single mode / neighboring pair

b

0.00\,

Total prepared modal amplitude

A

0.12\,

Inspected site along chain

s/L

50.00\,

Inspect acquired record

t/t_{end}

0.00\,\%

Click or drag along the rail below to pick a bead. Every setting also has a keyboard slider. Picking another bead or showing and hiding items keeps the cursor where it is; changing the starting state begins a new trial. The pure-mode buttons jump to exact times from the analytic solution. Fine step moves forward by five thousandths of the normalized time unit.

The masses are equal and normalized, the springs are identical ideal Hooke springs, the ends are fixed and every bead starts at rest. The beads move along the chain; this is not a model of a string vibrating sideways. Each normal mode evolves analytically, with no time stepping, and the energy computed from bead positions and velocities is found separately from the mode energies. Beats use the two highest neighboring modes. The reported period is how long the absolute value of the beat envelope takes to repeat, which need not be when the whole chain returns to the same state. Damping, sideways motion, spring mass, coil mechanics and collisions are left out. The preset amplitudes keep the ideal springs within the stated normalized strain range. The cursor shows five decimal places, but that does not mean experimental accuracy.

Start with the essentials

Focus question: Why can a chain of interacting masses still be split into as many independent simple motions as there are masses?
One-sentence intuition: In the right collective coordinates, a coupled system becomes a set of independent harmonic oscillators.

Core mathematical model

Discrete spring chain

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

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

Fixed-end normal frequencies

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

Each mode $p$ 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 misconceptionThe slow rise and fall of the amplitude means a new low-frequency mode has appeared.

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

Run the experiment

01
Find the lowest mode
Choose Mode 1, press First-mode quarter period and look at each bead.
What to observe: All masses move broadly in the same direction.
02
Count nodes
Compare Mode 2 with Mode 3.
What to observe: Higher mode number adds internal nodes and raises frequency.
03
Add two modes together
Choose Neighboring-mode beats.
What to observe: A slow envelope wraps the fast oscillation, yet the energy in each mode stays constant.