What you are looking at
A mass on a spring, given a rhythmic push F₀cos(ω_d t) — the gold arrow. The lower strip shows the motion
and the drive on one time axis; the right graph is the punchline: the
steady-state amplitude as a
function of drive frequency, with a gold dot marking where you are on the curve.
The equation of motion
Newton's second law with three horizontal forces — spring, damping, drive:
m·ẍ = −k·x − c·ẋ + F₀·cos(ω_d t)
After the transients die out the mass settles into oscillating
at the drive frequency with a fixed
amplitude:
A(ω) = F₀ / √( (k − mω²)² + (cω)² )
That denominator is smallest — and A largest — when ω ≈ ω₀ = √(k/m):
resonance.
Three regimes, one curve
Slow driving (ω ≪ ω₀): the spring simply stretches with the force, A ≈ F₀/k, motion in
phase with the push.
At resonance: every push arrives exactly in step with the velocity,
feeding energy in all cycle long; amplitude climbs until damping burns energy as fast as the drive supplies
it, A ≈ F₀/(cω₀) — the quality factor Q sets how tall and sharp the peak is.
Fast driving
(ω ≫ ω₀): inertia wins, the mass barely moves and lags the force by nearly 180°.
The RLC circuit is this exact system
Compare with the RLC resonance simulator: the equations map one-to-one — L↔m, R↔c, 1/C↔k, charge↔position,
drive voltage↔drive force. The current peak you tuned there is precisely this amplitude peak; a radio "tunes
in a station" by sliding its resonance peak onto the station's drive frequency.
Why engineers fear and love it
Loved: radios, musical instruments, MRI, quartz watches all live on sharp resonances. Feared: soldiers break
step on bridges, buildings are damped against earthquakes, and washing machines shudder as they pass their
spin-up resonance — a driven system near ω₀ with little damping stores enormous amplitude.
Things to try
Sweep the drive slowly through f₀ and watch the gold dot ride over the peak while the mass's swings balloon
and shrink. Halve the damping and the peak doubles and sharpens. Raise k and the whole peak slides to higher
frequency — retune the drive to chase it, exactly like turning a radio dial.