Driven Resonance: Amplitude, Damping and the Q Factor

A mass, a spring, a push

A damped mass on a spring, pushed by an oscillating external force, is one of the most-studied systems in physics because so many real things reduce to it: bridges in wind, buildings in earthquakes, atoms absorbing light, radio receivers tuned to a station. Newton's second law for the driven, damped oscillator is:

m·x'' + c·x' + k·x = F0·cos(ωt)

Here m is mass, c is the damping coefficient, k is the spring stiffness, and F0cos(ωt) is the external drive at angular frequency ω. Two frequencies matter: the system's own natural frequency ω0 = √(k/m), the rate it would oscillate at if left alone, and the driving frequency ω, which you control.

The steady-state amplitude curve

After any initial transient dies away, the mass settles into steady oscillation at the driving frequency ω, with an amplitude that depends on how close ω sits to ω0 and how much damping is present:

A(ω) = F0 / m·sqrt[(ω0² - ω²)² + (c·ω/m)²]

Sweep ω slowly from zero and A(ω) climbs, peaks near ω0, then falls back off — the resonance curve. For an undamped system the peak would be infinite exactly at ω0; real damping caps it at a finite value and shifts the peak very slightly below ω0.

What damping does to the peak

Damping's job is entirely about the peak — it caps its height and widens its base. Light damping gives a tall, narrow resonance peak: the system responds enormously to frequencies very close to ω0 and almost not at all elsewhere. Heavy damping gives a short, broad peak: the system responds moderately across a wide band of frequencies but never dramatically to any single one. This trade-off between a sharp, sensitive response and a broad, forgiving one is unavoidable — you cannot have both from the same physical system.

The Q factor quantifies it

The quality factor Q packages that trade-off into one number: Q = mω0/c (equivalently, Q ≈ ω0 divided by the width of the resonance peak at half its maximum power). High-Q systems — a tuning fork, a laser cavity, a well-made bell — ring for a long time at a very precise frequency and are extremely selective about what drives them. Low-Q systems — a car's suspension, a door closer — respond broadly and die out fast, which is exactly what you want when the goal is to absorb a shock rather than resonate with it. Radio tuning circuits are deliberately built high-Q so they pick one station cleanly out of a crowded spectrum; shock absorbers are deliberately built low-Q so a pothole does not set the car ringing.

Why resonance can be dangerous

Feeding a lightly-damped structure a drive near its natural frequency is how bridges have failed and machinery has torn itself apart: the amplitude does not grow without bound as it would in the idealized zero-damping case, but the finite peak a real structure settles at can still vastly exceed its static deflection under the same force, enough to exceed a material's elastic or fatigue limit. Engineers deliberately push a structure's natural frequencies away from expected driving frequencies (footsteps, wind gusts, motor vibration) or add damping specifically to blunt the resonance peak — tuned mass dampers in skyscrapers and the retrofit dampers added to London's Millennium Bridge after pedestrian-induced sway are both direct, real-world applications of this same A(ω) curve.

Frequently asked questions

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