The Damped Oscillator — Under, Over, and Critical

Source

Lecture: Classical mechanics — oscillations: the free damped oscillator, ẍ + 2αẋ + ω₀²x = 0.
Simulation: Spring-mass released from x₀ = 1 m with a live x–t trace; a one-mass explore mode plus a counterfactual three-lane settle race (α = 0.6 / 3 / 6 rad/s).
Concept: Damping speeds the return to rest only up to α = ω₀; past critical, the slow root α − √(α²−ω₀²) makes the mass crawl back.
Serves: Regime classification (underdamped / critical / overdamped); exact closed-form solutions in all three regimes; the non-monotonic settling-rate peak λ_max = ω₀ at α = ω₀.
Inquiry: Before dragging, predict which reaches rest first — the critical mass or the most heavily damped — then run the race and watch the decay-rate readout climb to its peak at critical and drop past it.
Misconception: "More damping always settles faster" — past critical the mass returns SLOWER; the fastest non-oscillatory return is at α = ω₀ exactly.
Decay-rate map: λ(α) = α − √(max(0, α² − ω₀²)) — the rate of the slowest surviving exponential. The white dot on the map tracks your slider’s α.

Pedagogy principles applied

ID	Principle	How it shows up here
CF	Counterfactual mode	✓ One mass vs ⚠ Settle race mode buttons; the naive model x = e^(−αt) is drawn exactly, dashed, against the true solution whenever α > ω₀ — visibly failing.
RV	Real values, honest units	SI throughout (m, s, rad/s); positions are the exact kernel solutions (x(1.2 s) = −0.488 / 0.126 / 0.411 at α = 0.6 / 3 / 6) — no hand-tuned decay.
RD	Readout discipline	Two numerics only (α and the slowest decay λ); settle-time stamps appear in-scene only when a lane actually reaches rest.
HS	One hero scene	Spring rig + strip chart own the viewport; the λ(α) map and equations live in collapsed / toggled panels.
PF	Prediction first, optional	The race enters paused at t = 0 with the prediction offered above the stage; Play is never gated and works in one click.
H2	Misconception structural	"Heavier stops soonest" exists only as a ⚠ prediction choice and as the subordinated dashed naive curve; the rebuttal appears only if chosen.
PK	Invariant / peak made visible	The λ readout + meter rises 1:1 with α to exactly 3.00 rad/s at the slider's green tick (α = ω₀), then falls — the whole lesson in one drag.
AH	Audit hook	window.__audit.at(α) returns the identical λ = α − √(max(0, α²−9)) formula the readout displays (no Lorentz γ — Galilean carve-out).
PL	Playable from load	Animation live on load, slider enabled, prediction answerable immediately; nothing is stage-locked.

The Damped Oscillator

Teach me

Speed

Guided Inquiry

1 · Predict — the settle race

Let’s open with a race. Damping α opposes a mass on a spring, with ω₀ = 3 rad/s. Three identical masses get the same release: light α = 0.60 · critical α = 3 · heavy α = 6. Which do you think is first to rest (inside the band and staying)?

2 · Run the race

Now let’s see how your prediction holds up. Settle race is on — press ▶ Play if needed. Critical hits the band first, while heavy still crawls. "More damping always settles faster" fails in plain sight.

3 · Watch the light case

Now let’s slow down and watch just one mass, with α = 0.60 rad/s (< ω₀ = 3). It’s released from x₀ = 1 m with v₀ = 0. What do you notice — does the trace go straight to rest, or cross x = 0 and ring before settling in the shaded ±2 cm band?

4 · Where ringing stops

Let’s turn the damping up: α is now on the green tick, α = ω₀ = 3 rad/s. Compared with the underdamped trace, what do you think has happened to the crossings of x = 0?

Confirmed. The last zero-crossing has just vanished — critical damping. Watch λ: it sits at its peak, 3.00 rad/s.

5 · Past the summit

Now let’s double it: α = 6 rad/s, twice critical. There’s no overshoot, but take a look at slowest decay λ. Against the critical case, do you think it has risen, fallen, or held?

λ: it has fallen below 3. More friction killed the ring and slowed the return.

6 · λ peaks at α = ω₀

Let’s put the whole story together. Back on one mass, underdamped (α = 0.60), watch it ring while λ = α. Then drag α across the green tick — λ climbs to ω₀ at critical, then falls as you go overdamped. Underdamped: more α kills ringing faster. Overdamped: more α makes the slow exponential crawl.

Listen

Damping control

▾

damping α α = 0.60 rad/s

UNDERDAMPED

Decay rate λ

▾

λ = 0.60 rad/s

Decay-rate map λ(α)

▾

Equation of motion (drawn from its exact solution)

Underdamped α < ω₀ — rings

Critically damped α = ω₀ — no overshoot, fastest

Overdamped α > ω₀ — two real decays

Slowest decay rate — the λ readout, peaked at critical

Settling time (2 % band)

The Physics

The damped oscillator · what this sim shows, the physics behind it, and every control

At a glance

A mass on a spring, with a dashpot adding friction, is released from rest at x₀ = 1 m. You set the friction and watch how, and how quickly, it returns to rest: the question behind every shock absorber.

What you'll learn

How one equation of motion produces different kinds of return to rest.
How the shape of the motion depends on comparing the damping α with ω₀.
What the decay rate λ measures, and how it depends on α.
How to tell which damping brings the mass to rest soonest.

The rig: a spring and a dashpot share the load. The mass starts 1 m above equilibrium, at rest.

What's on screen

The mass (violet block): its height is the displacement x.
The x–t trace: x against time, with a dot at the present moment.
Green band: ±2 cm around zero. A mass that enters it and stays is at rest (green ✓).
Dashed line: x = 0, the equilibrium position.
Settle race: three masses, α = 0.6 (sky), α = 3 (green) and your α (amber).
Sidebar: the regime badge, the decay rate λ, and the λ(α) map with a marker at your α.

See itRelease one mass with α = 0.60 rad/s. Watch for: the block and the head of the trace always at the same height. The trace is the block's height over time.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
One mass / Settle race	One mass on its own, or three dampings released together. Entering the race sets α to 6 for the third lane.	two scenes opens on One mass
damping α	The strength of the dashpot's drag. Every change restarts the release from x₀. In the race it sets the amber lane.	0–6 rad/s opens at 0.60
α = ω₀ (green tick)	Jumps α to exactly 3 rad/s, the value of ω₀.	button
Regime badge	Reads UNDERDAMPED, CRITICAL or OVERDAMPED, from comparing α with ω₀.	label
Decay rate λ	The rate of the slowest part of the motion that survives. Larger λ means the motion dies out sooner.	panel
Decay-rate map λ(α)	λ plotted against α over the slider's range, with a marker at your α.	panel
Time zoom	Scroll on the graph to widen or narrow the time axis.	0–8 s to 0–20 s
▶ Play / ⏸ Pause	Runs the release, or freezes it so you can read the trace.	top bar
Speed	Plays the motion faster or slower on screen. The physics stays the same.	0.25×–4×
↻ Reset	Restarts the mode you are in from its beginning.	top bar

Fixed: ω₀ = 3 rad/s (spring and mass), x₀ = 1 m released from rest, rest band ±2 cm (2% of x₀).

One equation, three motions

Per unit mass, the spring pulls back with ω₀²x and the dashpot drags with 2αẋ:

The equation of motion. The sim draws its exact solution; nothing is stepped numerically.

Which kind of motion you get depends on one comparison, α against ω₀ = 3 rad/s: underdamped (α < ω₀), critical (α = ω₀) or overdamped (α > ω₀). Below ω₀ the mass oscillates inside a shrinking envelope:

Underdamped: a cosine at ω_d times the envelope e^−αt.

The mass overshoots x = 0 and swings back, each swing smaller than the last.
Damping also slows the swing: ω_d = √(ω₀² − α²) is less than ω₀. At α = 0.6, ω_d = 2.94 rad/s, a period of 2.14 s against 2.09 s with no damping.

See itOne mass with light damping, α = 0.60 rad/s. Watch for: the trace crossing x = 0 again and again, each swing smaller than the one before.

Critical damping ends the ringing

As α rises towards ω₀, ω_d = √(ω₀² − α²) shrinks, so the swings get slower as well as smaller. At α = ω₀, ω_d = 0 and the oscillation is gone. This boundary is critical damping, the green tick on the slider:

Critical, α = ω₀: no cosine left, only a growing factor times a decay.

Above ω₀ both roots of the equation are real and negative, so the motion is a sum of two decaying exponentials:

Overdamped, α > ω₀: two decays, a fast one (r₂) and a slow one (r₁).

Key idea. Released from rest, a critically damped or overdamped mass never crosses x = 0. It only creeps towards it.

See itSet α exactly on the green tick, α = ω₀ = 3 rad/s. Watch for: the trace sinking into the green band without ever crossing the dashed x = 0 line.

The decay rate peaks at critical

Long after release the motion is ruled by its slowest exponential, x ∝ e^−λt. That rate λ is the sim's λ readout:

Below ω₀, λ is the envelope rate α. Above ω₀, λ is set by the slow root.

Below ω₀, more damping means faster decay: λ = α.
Above ω₀, λ = α − √(α² − ω₀²) falls as α grows: 3.00 rad/s at α = 3, 1.35 at α = 4, 0.80 at α = 6.
The peak, λ = ω₀ = 3 rad/s, is exactly at critical damping.

From the sim's own formulas: ω_d (blue) reaches zero at the same α where λ (violet) peaks.

This slider moves the sim's α too.

See itThe sim sweeps α from 0.6 to 6 rad/s with the λ(α) map open. Watch for: λ climbing to 3.00 at α = 3, then falling to 0.80 at α = 6.

Why heavy damping is slow

Same release, three dampings, from the exact solutions. Dots mark when each mass enters the ±2 cm band for good.

Overdamped motion is two exponentials. The fast one dies almost at once; the slow one, with rate λ, sets how long the return takes.
As α grows the slow rate shrinks, roughly as λ ≈ ω₀²/(2α) for α well above ω₀ (at α = 6 this gives 0.75 rad/s; the exact value is 0.80).
The picture: a strong dashpot resists motion at every speed, so the spring can only push the mass home slowly.
In the race, measured as the time to enter the ±2 cm band and stay: α = 3 rests at 1.9 s, α = 6 at 5.0 s, and α = 0.6 at 6.5 s, after swinging back and forth.

Too little damping is slow because the mass keeps swinging; too much is slow because it crawls.

See itRun the settle race: α = 0.6, 3 and 6 from the same release. Watch for: the green mass gets its ✓ first, near 1.9 s, then amber near 5.0 s, then sky near 6.5 s.

Where it matters

Engineers describe damping by the ratio ζ = α/ω₀: below 1 it rings, at 1 it is critical, above 1 it crawls.

Car suspension. Shock absorbers are usually set well below critical, often around ζ ≈ 0.2–0.4: the car rises and dips once or twice after a bump, which rides more softly than a stiffer setting.
Door closers. Set close to critical, so the door shuts quickly without slamming. With too much damping it creeps through the last few centimetres.
Analog meters. Needle instruments are damped a little below critical, around ζ ≈ 0.7, so the pointer reaches its reading quickly with only a tiny overshoot.

Go deeper: critical is not always first into the band

Critical damping gives the largest λ and the fastest return with no overshoot. That is not the same as the shortest time to enter a tolerance band.
In this sim, α = 2.35 rad/s (ζ ≈ 0.78) overshoots by 1.9 cm, which stays inside the ±2 cm band, and is at rest from 1.2 s, against 1.9 s for critical.
This is why instruments are often damped slightly below critical: a small overshoot inside the tolerance costs nothing.

Summary

In one line. α against ω₀ decides whether the mass rings, returns without overshoot, or crawls, and the decay rate λ is largest at critical damping, α = ω₀.

Key points

Underdamped (α < ω₀): oscillation at ω_d = √(ω₀² − α²), inside the envelope e^−αt.
Critical (α = ω₀): ω_d = 0, no zero crossing, and λ at its peak, ω₀.
Overdamped (α > ω₀): two exponentials; the slow one, λ = α − √(α² − ω₀²), falls as α grows.

Common mix-ups

"More damping always settles it sooner." Only up to α = ω₀. Beyond that the slow exponential gets slower.
"A damped mass rings at ω₀." It rings at ω_d, always below ω₀.
"Critically damped means it stops at once." It still decays, as (1 + ω₀t)e^−ω₀t, and here takes 1.9 s to settle into the band.

Things to try on your own

Set α = 0 and watch the mass swing without ever settling: λ reads 0.
Read λ at α = 1.5, 3 and 6, and check each against the formulas in section 05.
With One mass, note where the ✓ flag lands on the time axis at α = 3, then at α = 2.35.