Force Transmissibility⚠ unverified

Understanding the Parameters

• Frequency ratio $r = \omega/\omega_n$ — the controlling variable. The isolation strategy is to make the mount soft (low $\omega_n$) so that the disturbance frequency $\omega$ sits well above $\omega_n$ ($r \gg 1$).

• The three regions — $r < 1$: quasi-static, $TR \approx 1$. $r \approx 1$: resonance, $TR$ peaks (huge amplification if lightly damped). $r > \sqrt2$: isolation, $TR < 1$, falling as $\sim 1/r^2$.

• Damping's double role — damping reduces the resonant peak (good, protects the mount during run-up) but worsens high-frequency isolation (bad, the $(2\zeta r)^2$ numerator term limits the roll-off). Mount design trades these off; wire-rope and elastomeric mounts add damping deliberately for run-up survival.

• The $\sqrt2$ crossover — at $r = \sqrt2$, $TR = 1$ regardless of damping; this is the universal boundary between amplification and isolation.

Derivation (Approaching a Proof)

For a base-excited (or force-excited) SDOF system $m\ddot{x} + c\dot{x} + kx = $ (harmonic input), the steady-state response is found by assuming $x = X e^{i\omega t}$. The transmitted quantity passes through the spring and damper in parallel, so it has both a stiffness part ($k$) and a damping part ($c\omega$), giving a numerator $\sqrt{k^2 + (c\omega)^2}$. Dividing the transmitted amplitude by the input and normalising ($r = \omega/\omega_n$, $\zeta = c/2m\omega_n$) yields

$$TR = \frac{\sqrt{1 + (2\zeta r)^2}}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}.$$

The denominator $\sqrt{(1-r^2)^2 + (2\zeta r)^2}$ is the resonance term (it vanishes toward zero near $r=1$ for small $\zeta$, causing the peak); the numerator's $(2\zeta r)^2$ is the damper's transmission path. Setting $TR = 1$ gives $(1-r^2)^2 = 0$ only at $r=\sqrt2$... more precisely $1 + (2\zeta r)^2 = (1-r^2)^2 + (2\zeta r)^2 \Rightarrow (1-r^2)^2 = 1 \Rightarrow r = \sqrt2$ (taking the isolation root).

Dimensional check. $r$, $\zeta$ are dimensionless, so $TR = \sqrt{(\text{–})/(\text{–})}$ is dimensionless — a ratio, as required.

History and Development

Transmissibility theory is the foundation of vibration isolation, developed in the early 20th century for machinery mounting and refined for aircraft, vehicles, and precision instruments. The counter-intuitive results — that a mount amplifies below $r=\sqrt2$, that soft mounts isolate better, and that damping trades resonance protection for high-frequency isolation — are all read directly off this curve. That the force and motion transmissibilities share one formula (hence the duplicate cards) reflects a deep input–output symmetry of the SDOF system.

Related Concepts: Transmissibility, Vibration Isolation Efficiency, Magnification Factor, Natural Frequency Mass Spring, Damping Ratio, Damped Natural Frequency

Notes: Duplicate of Transmissibility (identical formula; force ≡ motion transmissibility). Isolation only for $r > \sqrt2$; $TR = 1$ at $r=\sqrt2$ for any $\zeta$. Damping cuts the resonant peak but worsens high-$r$ isolation. Roll-off $\sim 1/r^2$. Isolation efficiency $= 1 - TR$ (Vibration Isolation Efficiency).