#### Smallwood Algorithm Calculate acceleration response Rayleigh Distribution is a special case of Weibull distribution (shape factor = 2) Useful for fatigue analysis and estimating peak response ## Random Vibration ### Crest Factor #### Rayleigh Peak Response Formula Expected peak response to white noise, for large $T$ right term goes to zero (most cases) $ C_{n} = \sqrt{ 2 \ln (f_{n}T) } + \frac{0.5772}{\sqrt{ 2\ln(f_{n}T) }} $ ${} f_{n} =$ natural frequency (Hz) $T =$ duration (s) $0.5772$ is Euler's constant For worst case crest factor with probability below $\lambda_{\alpha}$ $ \lambda_{\alpha} = \left( \sqrt{ 2 \ln (f_{n}T) } + \frac{0.5772}{\sqrt{ 2\ln(f_{n}T) }} \right) \sqrt{\frac{-\ln(1-(1-\alpha)^{1/(n_{0}^{\dagger}T)})}{\ln(n_{0}^{\dagger}T)} } $ #### Power Spectral Density (PSD) #### Shock Response Spectrum (SRS) ![[Pasted image 20250608175017.png]] Acceleration time history → series of damped sine responses ![[Pasted image 20250608175118.png|450]] #### Vibration Response Spectrum (VRS) #### Damping ![[Pasted image 20250817234652.png|450]]