Magnification Factor⚠ unverified
Mechanical / Vibration Analysis · Compute the dynamic magnification factor of a forced oscillator
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| wn | wn | rad/s | 1.0 | Undamped natural frequency |
| omega | ω | rad/s | 1.0 | Excitation (forcing) frequency |
| zeta | ζ | — | 1.0 | Damping ratio (dimensionless) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | M | — | Dynamic magnification factor (dimensionless). Returns 0.0 when the denominator is non-positive |
The science & history
Understanding the Parameters
- $r = \omega/\omega_n$ — $r=1$ is undamped resonance; with damping the peak shifts slightly below 1.
- $\zeta$ — controls peak height ($M_{\max} \approx 1/(2\zeta\sqrt{1-\zeta^{2}})$).
- $M$ — $X = M(F_0/k)$ for harmonic force amplitude $F_0$.
Derivation (Approaching a Proof)
Particular solution of $m\ddot{x}+c\dot{x}+kx = F_0\sin\omega t$ yields complex receptance whose modulus, normalised by $1/k$, is $M(r,\zeta)$ as above.
History
The magnification (or dynamic amplification) factor is the standard forced-response chart in vibration textbooks.
Related Concepts: Transmissibility, Resonance Frequency damped, Damping Ratio, Natural Frequency mass-spring
Notes: Registry calculator vibration-magnification-factor (unverified). Base-fixed force excitation.