forced vibration and resonance
Free vibration in the last lesson died away once you let the mass go. This lesson keeps pushing. A playground swing makes the picture clear. Push at random moments and it hardly climbs, but push in time with the swing's own beat and it rises higher each time. The secret sits in that beat. Drive a system with a periodic force and the system ends up oscillating at the beat you impose. When that beat matches the natural frequency ωₙ, the push adds energy in the same direction as the motion on every cycle. The amplitude then grows large. The only thing that holds it back is damping. A small force, if it keeps the right time, can rock a bridge, and a voice alone can break a glass. The phenomenon in which the beats coincide is resonance, and it is the most important and the most dangerous idea in vibration.
Push the system with a periodic force F = F₀cos(ωt). At first a transient appears, still mixed with the free vibration of the last lesson, but once that dies the system settles into a steady oscillation. The key point is that the system does not swing at its own natural frequency ωₙ. It swings at the driving frequency ω that the push sets. The amplitude is the static deflection F₀k multiplied by a magnification M, and this M depends on the frequency ratio r = ωωₙ. When ω sits far from ωₙ the mass hardly moves. When ω approaches ωₙ the amplitude swells. The system moves most willingly when it is pushed at the beat it was born with. Off the beat, the same force barely moves the mass.
Plot that magnification M against the frequency ratio r and the resonance curve appears. It is M = 1√((1−r²)²+(2ζr)²), and the curve splits into three zones. When r is small, a slow push, M ≈ 1. The mass follows the force almost as it is and deflects only by F₀k, moving as if the force were static. When r is near 1 the denominator falls almost to 0 and M climbs into a sharp peak. That is resonance. When r is large, a fast push, M → 0, so the mass is too sluggish to follow and hardly responds. The peak at r ≈ 1 is the heart of the curve. Push in time with the natural frequency and the response grows by several times.
What stops the peak from climbing without end is damping. At resonance (r = 1) the (1−r²) term becomes 0, so the only term holding the amplitude is 2ζr, the damping. Change ζ and a light damping makes the peak tall, sharp, and dangerous, while a heavy damping makes it low and blunt. The peak height is roughly 1(2ζ), and that quantity is called the quality factor Q. A bell or a wine glass with almost no damping (high Q) rings in a sharp, pure resonance, and a well-damped machine mount (low Q) hardly responds. Engineers add damping not to move the natural frequency but to tame the resonance peak.
The story has a second half, and that half is phase. The response always lags the driving force by an angle φ. Below resonance (r < 1) the lag is small, so the mass and the force move almost together. Right at resonance (r = 1) the lag is exactly 90°, which is 14 of a cycle. There the force always pushes in the direction the mass is already going, so energy enters efficiently and the amplitude grows. Above resonance (r > 1) the lag approaches 180°, and the mass moves against the force. Almost nothing is then handed across. Watch the drive point and the response point: they move together, then slip by 14 of a cycle, then move in opposite directions.
Resonance is everywhere, for good and for ill. The Tacoma Narrows bridge (1940) twisted itself apart once the wind fed energy at the bridge's natural frequency, and it became the textbook warning about destructive resonance. A singer can break a wine glass by holding the note that matches the glass's natural frequency. Soldiers break step on a bridge so the marching beat never locks onto the bridge's natural frequency. Resonance is also useful. Radio tuning sets 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 many, is the same physics with the opposite intention.