Resonance — When a Small Push Becomes a Big Shake
Last lesson's free vibration died out once you let go. This time we keep pushing. Think of a playground swing. Push at random moments and it hardly climbs. But push in time with the swing's own beat and it goes higher and higher. The whole secret is right there. Drive a system with a periodic force and it ends up oscillating at the beat you impose. If that beat happens to match the system's natural frequency ωₙ, each push adds energy in the same direction the motion is already going, and the amplitude builds enormously. What holds it back? Only damping. A small force, kept in rhythm, can shake a bridge apart or shatter a glass with a voice. That matching of beats is resonance, the most important and most dangerous idea in vibration.
Push the system with a periodic force F = F₀cos(ωt). At first there is a brief transient that still carries the free vibration from the last lesson, but once that dies away the system settles into a steady state. Here is the key point: the system oscillates not at its own natural frequency ωₙ but at the driving frequency ω that you impose. It dances to the beat you give it. The amplitude is the static deflection F₀k times a magnification factor M, and that M depends on the frequency ratio r = ωωₙ. Move ω: far from ωₙ the mass barely moves, but as ω approaches ωₙ the amplitude swells. The system moves most readily when you push at its natural beat.
Plot that magnification M against the frequency ratio r and you get the resonance curve, M = 1√((1−r²)²+(2ζr)²). It splits into three zones. At small r (a slow push), M ≈ 1: the mass simply follows the force, deflecting by F₀k as if the force were static. Near r = 1 the denominator nearly vanishes and M shoots up into a sharp peak — this is resonance. At large r (a fast push), M → 0: the mass is too sluggish to keep up and barely responds. The peak at r ≈ 1 is the heart of it: push at the natural frequency and the response is amplified many times over.
What keeps the peak from going infinite? Damping. At resonance (r = 1) the (1−r²) term is zero, so the only thing holding the amplitude back is the 2ζr term, the damping. Move ζ: light damping gives a tall, sharp, dangerous peak; heavy damping gives a low, broad one. The peak height is roughly 1(2ζ), which is called the quality factor Q. A barely damped bell or wine glass (high Q) rings with a sharp, pure resonance, while a well-damped machine mount (low Q) hardly responds. This is exactly why engineers add damping: not to change the natural frequency, but to tame the resonance peak.
There is a second half to the story: phase. The response always lags the driving force by an angle φ. Below resonance (r < 1) the lag is small, so mass and force move almost together. Right at resonance (r = 1) the lag is exactly 90°, a quarter cycle: here the force is always pushing in the direction the mass is already moving, which is exactly why energy pours in so efficiently and the amplitude grows. Above resonance (r > 1) the lag approaches 180°: the mass moves opposite to the force, fighting it, so little gets through. Watch the drive dot and the response dot: they shift from moving together, to a quarter-cycle apart, to moving in opposite directions.
Resonance is everywhere, for good and ill. The Tacoma Narrows bridge (1940) twisted itself apart once the wind fed energy at the bridge's natural frequency — the textbook warning about destructive resonance. A singer can shatter a wine glass by holding the note that matches the glass's natural frequency. Soldiers break step when crossing a bridge so their marching rhythm never locks onto the natural frequency. But resonance is also useful: tuning a radio means adjusting a circuit's natural frequency until it resonates with one station and ignores the rest — resonance as a filter that picks one signal out of thousands. Same physics, opposite intentions.