Magnification Factor of a Forced Vibration

Apply a force slowly to a spring and it deflects by $F/k$. Apply the same force oscillating, and the deflection can be very much larger — or very much smaller. The magnification factor is the ratio between the two, and it is the amplitude side of the same curve transmissibility describes: what the MACHINE does, rather than what the floor receives.

Three regions again, and each has a plain-language reading. Well below resonance, $M \approx 1$: the structure follows the force quasi-statically, and treating a slow-moving load as a static one is legitimate. Near $r = 1$, the first two terms cancel and the whole answer collapses to $M = 1/(2\zeta)$ — a structure at $\zeta = 0.02$ amplifies its own static deflection twenty-five times. That is precisely why resonance is dangerous: the force has not grown at all, but the same force now produces motion tens of times larger. Above resonance, $M$ falls as $1/r^2$ and approaches zero, because the mass simply cannot follow the force any more.

A detail worth having: the peak does not sit exactly at $r = 1$. It sits at $r = \sqrt{1-2\zeta^2}$, marginally below, and vanishes altogether once $\zeta > 1/\sqrt{2} \approx 0.707$ — past that damping there is no resonant peak at all and the response falls monotonically from the start. For the light damping found in machinery the distinction is academic, but it is why a measured peak frequency, a damped natural frequency and an undamped natural frequency are three slightly different numbers that people routinely treat as one.

Two extensions are worth knowing. The half-power method inverts this curve to measure damping: find the two frequencies either side of the peak where the response has fallen to $1/\sqrt{2}$ of its maximum, and $\zeta \approx (f_2 - f_1)/(2f_n)$ — the same relation as bandwidth and Q in an electrical resonant circuit, where $Q \approx 1/(2\zeta)$. And this equation covers steady harmonic forcing only. A load applied suddenly and held is a different problem entirely: a step load reaches a magnification of 2 regardless of damping, which is where the factor of two in impact and drop-load design comes from, and a true impact can go higher still.