Calculator · Dynamics & Control
Vibration Isolation & Transmissibility Calculator
Transmissibility of a single-degree-of-freedom mount at a given excitation frequency, with an explicit statement of which ratio each number is.
Mount and excitation
Results
| Natural frequency ωₙ | 5 rad/s |
|---|---|
| Damping ratio ζ | 0.15 |
| Frequency ratio r = ω/ωₙ | 3 |
| Absolute displacement transmissibility |X/Y| | 0.1671 |
| Force transmissibility |F_T/F₀| | 0.1671 |
| Relative displacement ratio |Z/Y| | 1.118 |
| Isolation boundary ω = √2·ωₙ | 7.071 rad/s |
Verdict at this frequency: isolation. The transmitted amplitude is smaller than the excitation amplitude: vibration isolation is achieved at this frequency. This is a transmission statement only, not a general safety statement.
Assumptions
- Single degree of freedom with linear stiffness and linear viscous damping, excited harmonically at one steady frequency.
- Steady state only: the start-up transient is excluded. Open the Vibrations Lab to see the transient.
- The relative displacement ratio |Z/Y| is the spring/damper stroke, NOT a transmissibility, and the r > √2 boundary does not apply to it.
- Isolation here is a transmitted-amplitude statement at one frequency only; it is not a statement about broadband excitation, shock, fatigue or structural safety.
Technical reference
m·ẍ + c·(ẋ − ẏ) + k·(x − y) = 0 (base excitation, no external force)
r = ω/ωₙ
T_x = |X/Y| = √(1+(2ζr)²) / √((1−r²)²+(2ζr)²) (absolute displacement / force transmissibility)
|Z/Y| = r² / √((1−r²)²+(2ζr)²) (relative displacement ratio, spring/damper stroke)
- y
- base (support) displacement[m]
- x
- absolute mass displacement[m]
- z = x − y
- relative displacement (mount stroke)[m]
- ω
- excitation angular frequency[rad/s]
- r
- frequency ratio ω/ωₙ[–]
- ζ
- damping ratio[–]
Under this base-excitation model, T_x and the (algebraically identical) force transmissibility T_F drop below 1 only for r > √2; that boundary does NOT apply to the relative displacement ratio |Z/Y|, which is a stroke, not a transmissibility. The verdict above is always taken from the computed ratio at the entered frequency, never from the r > √2 rule alone.
Steady-state, single frequency only — no transient, broadband or shock excitation.