Undriven Damped Harmonic Oscillator

The discriminant & three damping environments — how γ, m, and k together determine the nature of decay present in the system

The Equation (no driving force)

m (d²x/dt²) + γ (dx/dt) + k x = 0

Same structure as the driven oscillator — but the right-hand side is 0, not F₀cos(ωt)

How this differs from the driven oscillator: The only change is the right-hand side. Previously it was F₀cos(ωt) — an external force constantly pushing the system. Here it is 0, meaning no external input at all. We've removed the engine and are now watching the system coast to a stop purely under the influence of its own damping and spring forces.

What each symbol means

m, γ, k, and x carry the same meaning as in driven systems — but here their ratios to each other determine everything about the system's behavior.

Lowercase m

Mass

How heavy the object is — how much it resists acceleration. A larger m makes the system more sluggish to respond. It appears in the discriminant as the bottom of a fraction, so larger mass reduces Δ.

Greek letter gamma

Damping coefficient

Friction / resistance — the force that bleeds energy out of the system. This is the key player in the discriminant: Δ = γ² − 4mk. More damping (larger γ) drives Δ upward, tipping the system toward overdamped.

kg/s

Lowercase k (in the term kx)

Spring constant (stiffness)

The coefficient right in front of x in the term kx. It's the same spring stiffness from the driven oscillator, just without a driving force keeping things going. Larger k drives Δ downward, pushing the system toward underdamped (more springy).

N/m

Lowercase x

Displacement (what we solve for)

How far the object is from its rest position at any instant. This is still what we're trying to find — how does x change over time as the system winds down? The shape of that decay (oscillatory, smooth, sluggish) is exactly what the discriminant predicts.

m (meters)

= 0

Right-hand side equals zero

No external driving force

The zero on the right means the system is left entirely to itself — no external energy is being pumped in. Whatever energy was in the system at the start is all there is.

Called a "homogeneous" equation

What the discriminant is — and where it comes from

The discriminant isn't a term that is present in the equation. It's a value calculated from the equation's three coefficients.

The discriminant Δ

Δ = γ² − 4 · m · k

This formula comes from high-school algebra. In order to solve the equation of motion, we need to solve a quadratic equation — and the quadratic formula always contains the expression b² − 4ac under a square root. In this equation, a = m, b = γ, c = k, so b² − 4ac becomes γ² − 4mk.

The square root of Δ is what dictates the different behaviors in the three environments: if Δ is negative, then taking its square root produces imaginary numbers — and imaginary exponents produce oscillations. If Δ is positive then we observe two real solutions that just decay. If Δ is exactly zero, there's only one clean solution.

Quadratic formula slot	Standard algebra (ax² + bx + c = 0)	Our oscillator
a	coefficient of x²	m (mass)
b	coefficient of x	γ (damping)
c	constant term	k (stiffness)
Δ = b²−4ac	discriminant	γ² − 4mk

The three damping environments

Computing Δ in damped harmonic oscillator systems can tell us a lot about the physical characteristics of the environment in which our system resides.

Δ < 0

Underdamped

Not enough friction to stop oscillation. The system bounces back and forth, but each swing is smaller than the last. The amplitude decays exponentially while the oscillation continues.

Analogy: a plucked guitar string, or a car with worn-out shock absorbers that keeps bouncing after a bump.

Δ = 0

Critically damped

Friction is perfectly tuned — just enough to prevent oscillation without being so heavy that it slows the return. This is the fastest possible return to rest without any overshoot.

Analogy: a well-designed car door that swings shut and stops exactly in place — no bounce, no sluggishness.

Δ > 0

Overdamped

Too much friction. The system creeps back to rest without oscillating, but much more slowly than critical damping. The excess damping fights the return as much as the disturbance did.

Analogy: pushing a door underwater — no bouncing, but slow to close.

What the displacement x(t) looks like over time in each environment

All three curves start at the same displaced position and must eventually reach the rest line (x = 0). The discriminant predicts which path they take.

Worked example — three spring systems

We fix m = 1 kg and k = 16 N/m and vary only γ across three scenarios to demonstrate each damping environment. This shows how a single parameter change — the damping — steers the entire behavior.

Find the critical damping threshold — what γ would make Δ = 0?

It's useful to first calculate the γ value that sits right on the boundary. Set Δ = 0 and solve for γ.

Δ = 0 → γ² = 4 · m · k
γ² = 4 × 1 × 16 = 64
γ_critical = √64 = 8 kg/s

💡 This is the "Goldilocks" value. Less than 8 → underdamped. More than 8 → overdamped. Exactly 8 → perfect.

Scenario A: Underdamped — γ = 2 kg/s (light friction)

Δ = γ² − 4 · m · k
= 2² − 4 × 1 × 16
= 4 − 64
= −60

Δ < 0 → UNDERDAMPED (system will oscillate and decay)

💡 γ² = 4 is tiny compared to 4mk = 64. The spring stiffness hugely dominates the damping — the system wants to keep bouncing far more than the damping wants to stop it.

Scenario B: Critically damped — γ = 8 kg/s (perfect friction)

Δ = 8² − 4 × 1 × 16
= 64 − 64
= 0

Δ = 0 → CRITICALLY DAMPED (fastest return, no oscillation)

💡 Damping and stiffness are perfectly balanced. Engineers designing door closers, camera shutters, and instrument needles aim for exactly this.

Scenario C: Overdamped — γ = 20 kg/s (heavy friction)

Δ = 20² − 4 × 1 × 16
= 400 − 64
= +336

Δ > 0 → OVERDAMPED (slow creep back to rest, no oscillation)

💡 γ² = 400 crushes 4mk = 64. So much friction that the spring barely gets a say — the object just oozes back to rest.

Summary — what each result means physically

Δ = −60

Underdamped. Spring dominates — system bounces. Displacement oscillates while its amplitude shrinks. Good for musical instruments and suspension bridges (up to a point). Bad for a car's shock absorbers.

Δ = 0

Critically damped. Perfectly balanced — the system returns to rest as quickly as possible without overshooting. The engineering ideal for anything that needs to settle fast: instruments, door mechanisms, vehicle suspension.

Δ = +336

Overdamped. Friction dominates — system crawls back to rest without oscillating, but more slowly than critical damping. Useful when overshooting is catastrophic: galvanometer needles, some hydraulic systems.