Dynamics and Vibration Formulae

SDOF System

Natural frequency (angular):
  ω_n  =  √(k / m)

Natural frequency (Hz):
  f_n  =  (1 / 2π) · √(k / m)

Damping ratio:
  ζ  =  c / (2 · √(k · m))  =  c / c_cr

Critical damping:
  c_cr  =  2 · √(k · m)

Damped natural frequency:
  ω_d  =  ω_n · √(1 − ζ²)   (ζ < 1)

Dynamic Amplification (SDOF, harmonic excitation)

Frequency ratio:
  r  =  ω / ω_n  =  f / f_n

Dynamic magnification factor:
         1
  DMF  =  ─────────────────────
          √[(1 − r²)² + (2ζr)²]

At resonance (r = 1):
  DMF  =  1 / (2ζ)  =  Q

Logarithmic decrement:
  δ  =  (1/n) · ln(x_i / x_{i+n})  =  2πζ / √(1 − ζ²)
  (≈ 2πζ for small ζ)

Random Vibration

PSD definition:
  S(f)  =  lim [G(f) / Δf]    (units: g²/Hz or (m/s²)²/Hz)

Overall RMS from PSD:
  σ_RMS  =  √[ ∫ S(f) df ]

3-sigma stress (Gaussian):
  σ_3σ  =  3 · σ_RMS

Margin of safety:
  MS  =  (σ_allow / σ_3σ) − 1

Modal Effective Mass

Modal effective mass (mode j, direction i):
  M_eff,j,i  =  (L_j,i)² / M_j

  where L_j,i  =  φ_j^T · M · e_i  (participation factor)
         M_j   =  φ_j^T · M · φ_j  (generalised mass)
         e_i   =  influence vector (unit displacement in direction i)

Cumulative effective mass:
  Σ M_eff,j,i  →  M_total   (as sufficient modes are included)

Rule: include modes until Σ M_eff > 90% of total mass in the excitation direction.

Variables

Notes and Limitations

• SDOF formulae apply to single-mode response — for multi-mode response, use modal superposition
• PSD-based RMS assumes a stationary Gaussian random process
• The 3-sigma criterion (99.7% of peaks) may be exceeded for narrowband response — use the exact peak factor
• Damping ratio ζ is typically 0.01-0.05 for aerospace metallic structures, 0.03-0.08 for bolted/joint-rich structures

Related Knowledge

See the Dynamics, Vibration & Shock Knowledge category for modal theory and random vibration. See How to Perform a Modal Analysis and related guides.

• How to Perform a Modal Analysis
• How to Check Modal Effective Mass
• How to Perform a Random Vibration Analysis
• How to Interpret a PSD
• How to Calculate the RMS Response from Random Vibration
• How to Choose Structural Damping