M065 · Normal modes

Coupled Oscillators

Release a fixed-end longitudinal spring chain in one sine mode or two neighboring high modes. Inspect any bead, compare its two spring forces with acceleration, and follow displacement contributions and an independent coordinate-space energy ledger. The three-dimensional apparatus, exact event cursor and plots use the same analytic motion.

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 $0$

PREPARE / INSPECT / VERIFY

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 forces share a fixed trial scale.Inspected velocityBlue vector, on a separate 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 the rail below to select a bead. All edits have keyboard sliders. Changing the inspected bead or visibility preserves the cursor; changing the prepared release starts a new trial. Pure-mode actions seek analytic times. Fine step advances by five thousandths of normalized time.

Normalized equal masses and identical ideal Hooke springs, with fixed endpoints and zero initial velocity. Motion is longitudinal, not a transverse string model. Normal modes evolve analytically without time integration; coordinate-space energy is computed separately from modal energy. Beats use the two highest adjacent modes; the reported period is the recurrence of the absolute interference envelope, not necessarily recurrence of the complete chain state. No damping, transverse motion, spring mass, coil mechanics or collisions. Prepared amplitudes keep the ideal springs within the stated normalized strain range. Five-decimal cursor display is not experimental accuracy.

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

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 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

01
Identify the lowest mode
Choose Mode 1, seek its quarter period, and inspect individual beads.
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
Superpose two modes
Choose Beats.
What to observe: A slow envelope surrounds fast oscillation while each modal energy remains constant.