Indexxd: Dimensional Collapse in a Coupled-Oscillator Chain

Chain & Collapse

Full-chain DOFs (N) 10

Collapsed DOFs (K, far end) 5

Spring stiffness k 80

Damping c 0.30

Live Readouts

Effective DOFs

6 / 10

Compute saved

40%

Collapse error (RMS)

0.000

System energy

0.00

How it works

The full model is N point masses u₁…u_N in series between two fixed walls, each linked to its neighbours by springs of stiffness k and damped by c:

m·üᵢ = k(u_{i+1} − uᵢ) − k(uᵢ − u_{i−1}) − c·u̇ᵢ

Every DOF gets an index = its distance (in links) from the excited wall — the article's "indices assigned to each dimension, representing its relative importance." Low index = close to the excitation = dominant; high index = far away = safe to collapse.

The farthest K DOFs are replaced by a single lumped mass (mass = K·m) connected to the last kept mass by stiffness k and to the far wall by the series-reduced stiffness k/K — a real reduced-order model, not just a visual trick. Its ground-truth target, computed straight from the full simulation, is the article's weighted collapse formula:

u_collapsed = Σ(uⱼ · 1/indexⱼ) / Σ(1/indexⱼ),  j ∈ collapsed group

The collapse error readout is the live RMS gap between that target and the reduced model's actual lumped-mass position — i.e. how much accuracy the dimensional reduction is costing you right now, for the compute you're saving.

N — total DOFs in the full ground-truth chain.
K — how many of the farthest (highest-index) DOFs get merged into one.
Excite — kicks the wall-adjacent mass with a velocity impulse.
Continuous drive — keeps forcing DOF 1 sinusoidally so both models stay in steady oscillation.

Full chain (ground truth) Reduced chain (kept DOFs) Lumped collapsed DOF

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