No damping at all: oscillates forever at its natural frequency. A useful ideal; real systems always have some.
The common case: overshoots and oscillates, each swing smaller than the last. Bells, suspensions and tuned loops live here.
Returns to rest as fast as possible without overshooting: door closers, instrument needles.
So much damping the system crawls back, never oscillating: slower than critical. Past the sweet spot, more damping hurts.
The fastest response that keeps overshoot small (about 4%). Below it you get faster but bouncier; above, smoother but more sluggish. At ζ = 1 the two roots coincide and the response decays without crossing zero: the boundary between the oscillating and the sluggish worlds.
No: that is the surprise. Past critical, extra damping makes the system slower to return because it is fighting its own motion. Fastest settling is right around critical damping.
A car suspension, a building swaying in wind, an RLC circuit, the closed loop of a tuned controller. Two numbers decide the character: the natural frequency ωn (how fast it wants to oscillate) and the damping ratio ζ (how quickly the oscillations die away).
Two masses in series make a genuine 4-state coupled system with two natural frequencies (modes) instead of one. Pluck mass 1 and energy sloshes into mass 2: a simple model for a car’s body on suspension on tyre, or two floors of a building joined by a soft joint.
It assumes ideal viscous damping, a massless linear spring and small motion. Real systems add friction, nonlinear stiffness and more: treat it as the clean ideal, not the whole story.
It's a common design target — the fastest response that keeps overshoot small (about 4%). Below it you get faster but bouncier; above it, smoother but more sluggish.
At ζ = 1 the two roots of the system coincide and the response decays without crossing zero. It's the boundary between the oscillating (underdamped) and sluggish (overdamped) worlds.
No — that's the surprise. Past critical, extra damping makes the system slower to return, because it's now fighting its own motion. Fastest settling is right around critical damping.
It assumes ideal viscous damping, a massless linear spring, and small motion. Real systems add friction, nonlinear stiffness, and other effects — treat this as the clean ideal, not the whole story.
Two masses in series creates a genuine 4-state coupled system with two natural frequencies (modes) instead of one. Pluck mass 1 and energy sloshes back and forth into mass 2 — a simple model for anything from a car's body-on-suspension-on-tyre to two floors of a building connected by a soft joint.
Releases in which this page changed, newest last. Derived from the archived copy of every release, not from notes written afterwards — so it reflects what actually shipped. Site-wide passes are left out; they are in the full changelog.