## SDOF System Single degree of freedom system $m\ddot{q}(t) + c\dot{q}(t)+kq(t)=f(t)$ With initial conditions defined as $q(0)=q_{0}$ $\dot{q}(0)=\dot{q}_{0}$ ##### Free Vibration Natural harmonic oscillation from some initial displacement or velocity Governing equation: $m\ddot{q}(t) + c\dot{q}(t)+kq(t)=f(t)$ or $\ddot{q}+2\zeta\omega\dot{q} + \omega^{2}q = k\omega^{2}f(t)$ **Undamped Case** $q(t) = q_{0}\cos\omega t+\frac{\dot{q_{0}}}{\omega}\sin\omega t$ or $q(t) = Q\cos(\omega t-\theta)$ where $Q = \sqrt{ q_{0}^{2}+\left( \frac{\dot{q}}{\omega} \right)^{2} }$ and $\theta = \arctan\left( \frac{\dot{q_{0}}/(\omega Q)}{q_{0}/Q} \right)$ $\omega=\sqrt{ \frac{k}{m} }$ is the undamped natural frequency $2\zeta\omega = \frac{c}{m}$ $\frac{\zeta}{\sqrt{ 1-\zeta^{2} }}= \frac{1}{2\pi n}\ln \frac{Q_{i}}{Q_{i+n}} \approx \zeta$ for small $\zeta$ ##### Forced Vibration Excitation Sources 1. Base Excitation 2. Applied force or pressure 3. Initial displacement or velocity 4. Self-excited vibration, dynamic instability, flutter, POGO #### 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 whiite 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)} } $ #### Multiple Degree of Freedom Systems #### 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]]