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 — 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. Its ground-truth target, computed straight from the full simulation, is the 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 — how much accuracy the dimensional reduction costs for the compute it saves. This 2D view draws the two chains stacked, with vertical offset showing each mass's displacement.
- 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.