Vibration & Modal — Cheat Sheet
Natural frequency, damping, resonance, transmissibility and beam/rotor mode frequencies.
Free vibration (SDOF)
| Natural frequency | ωₙ = √(k/m) [rad/s] |
| In hertz | fₙ = ωₙ / 2π |
| Period | T = 1 / fₙ |
| From static deflection | fₙ ≈ 0.5·√(g / δ_st) |
| Torsional | ωₙ = √(k_t / J) |
Damping
| Damping ratio | ζ = c / c_c = c / (2√(km)) |
| Critical damping | c_c = 2√(km) = 2mωₙ |
| Damped frequency | ω_d = ωₙ√(1−ζ²) |
| Log decrement | δ = ln(x₁/x₂) = 2πζ / √(1−ζ²) |
Forced vibration & resonance
| Frequency ratio | r = ω / ωₙ |
| Resonance | r = 1 → amplify by 1/(2ζ) |
| Magnification | M = 1 / √[(1−r²)² + (2ζr)²] |
| Transmissibility | TR = √[1+(2ζr)²] / √[(1−r²)²+(2ζr)²] |
| Isolation region | r > √2 |
Continuous systems & modes
| Cantilever, 1st mode | fₙ = (3.516/2π)·√(EI / mL⁴) |
| Simply supported | fₙ = (π/2)·√(EI / mL⁴) |
| Axial rod | fₙ = (n/2L)·√(E/ρ) |
| String | fₙ = (n/2L)·√(T/µ) |
| Note | m = mass per unit length |
Rotating machinery
| Critical speed | N_c [rpm] = 60·fₙ |
| Rotating unbalance force | F = m·e·ω² |
| Rule of thumb | run well below/above ωₙ; pass resonance quickly |
Formula reference — verify against the governing standard. Not a substitute for engineering judgment.