Force Transmissibility

Transmissibility is the number isolation is judged by: the fraction of a machine's shaking force that reaches the floor through its mounts. Its shape is governed entirely by two dimensionless quantities — the frequency ratio $r = f/f_n$, the forcing frequency divided by the mount's natural frequency, and the damping ratio $\zeta$. Nothing else about the machine appears.

The curve has three regions and one crossing point. Below $r = 1$, transmissibility is greater than 1 and rising: the mount amplifies. At $r = 1$ it peaks, spectacularly if the damping is light. Between 1 and $\sqrt{2}$ it is falling but still above 1 — still amplifying. And at exactly $r = \sqrt{2} = 1.414$, transmissibility is exactly 1 for every damping ratio there is, because at $r^2 = 2$ the term $(1-r^2)^2$ equals 1 and the damping term appears identically above and below the line. Only past that crossover does the mount begin to do its job.

This is where the classic mistake happens, and it happens constantly. An isolator is chosen on instinct — softer must be better — without anyone computing where the running speed falls against the mount's natural frequency. If the ratio lands under $\sqrt{2}$, the installation is worse than bolting the machine down solid, and the complaint that follows ("we put isolators in and it got louder") is exactly what the equation predicts. The working rule is $r \geq 3$, which gives roughly 90% isolation at light damping and leaves margin for speed variation, for the mount stiffening as it ages, and for the second and third harmonics that ride along with the running speed. Variable-speed drives make this harder rather than easier: a machine that sweeps its speed has to pass through resonance every start, and a mount sized for full speed may sit below $\sqrt{2}$ at half speed.

Now the part that reads like a contradiction. Damping helps at resonance and HURTS above it. In the isolation region, more damping means MORE force transmitted, because the dashpot is a second path to ground that the spring by itself did not provide — the numerator $1 + (2\zeta r)^2$ grows with $\zeta$ faster than the denominator does out there. But damping is what limits the peak the machine passes through on every start-up and coast-down. So isolator selection is a trade-off and not an optimisation: a lightly damped mount isolates beautifully at speed and shakes hard on run-up; a heavily damped one survives run-up and leaks vibration all day. Which one is right depends on how often the machine starts, and no product resolves the tension.