Free Vibration

Damped, Single DoF Motion

• 
• $$\begin{array}{rl}y& =e^{-\zeta {\omega }_{n}t}\left(c_1\cos \left({\omega }_{d}t\right)+c_2\sin \left({\omega }_{d}t\right)\right)\\ & =e^{-\zeta {\omega }_{n}t}C\left(\sin \left({\omega }_{d}t\right)\right)\end{array}$$
• $$\begin{array}{rl}{Y}_0& =e^{-\zeta {\omega }_{n}t}C\\ \log \left(Y_0\right)& =-\zeta {\omega }_{n}t+\log \left(C\right)\end{array}$$
• 
• $$\begin{array}{rl}\log \left(Y_1\right)& =-\zeta {\omega }_n{t}_1+\log \left(C\right)\\ \log \left(Y_2\right)& =-\zeta {\omega }_n{t}_2+\log \left(C\right)\end{array}$$
• $\boxed{1}-\boxed{2}$:
  • $$\begin{array}{rl}\log \left(Y_1\right)-\log \left(Y_2\right)& =-\zeta {\omega }_n\left(t_1-t_2\right)\\ \log \left(\frac{Y_1}{Y_2}\right)& =\zeta {\omega }_n\left(t_2-t_1\right)\\ & =\zeta \frac{{\omega }_d}{\sqrt{1-{\zeta }^2}}\left(nT\right)\end{array}$$
  • $$\begin{array}{rl}\delta & =\log \left(\frac{Y_1}{Y_2}\right)\\ {\omega }_{d}T& =2\pi \\ ⟹& \boxed{\zeta =\frac{\delta }{\sqrt{{\left(2\pi n\right)}^2+{\delta }^2}}}\end{array}$$

EXAMPLE

• 
• Knowns:
  • $$\begin{array}{rl}{y}_{\text{static}}& =5.08\times 10^{-3}\left[\mathrm{m}\right]\\ mg& =267\left[\mathrm{N}\right]\end{array}$$
• Static:
  • $$\begin{array}{rl}mg& =k{y}_{\text{static}}\\ ⟹k& =\frac{mg}{y_{\text{static}}}\\ & =5.256\times 10^4\left[\frac{\mathrm{N}}{\mathrm{m}}\right]\end{array}$$
• Cut the line:
  • Beam begins oscillating
  • 
• Knowns:
  • $$\begin{array}{rl}{t}_1& =0\left[\mathrm{s}\right]\\ t_2& =1.5\left[\mathrm{s}\right]\\ \Delta t& =3T\\ \mathrm{y}\left(t_2\right)& =2.54\times 10^{-3}\left[\mathrm{m}\right]\end{array}$$
• Oscillation:
  • $$\begin{array}{rl}\delta & =\log \left(\frac{Y_1}{Y_2}\right)\\ & =\log \left(2\right)\\ \zeta & =\frac{\delta }{\sqrt{6{\pi }^2+{\delta }^2}}\\ & =0.0367\\ \Delta t& =3T\\ ⟹T& =\frac{\Delta t}{3}\\ & =0.5\left[\mathrm{s}\right]\\ {\omega }_d& =\frac{2\pi }{T}\\ {\omega }_n& =\sqrt{\frac{k}{m}}\\ ⟹m& =\frac{k}{{\omega }_n^2}\\ & =332.8\left[\mathrm{kg}\right]\\ 2\zeta {\omega }_n& =\frac{c}{m}\\ ⟹c& =2\zeta {\omega }_{n}m\\ & =307\left[\frac{\mathrm{N}-\mathrm{s}}{\mathrm{m}}\right]\end{array}$$
• Energy:
  • $$\begin{array}{rl}E& =\frac{1}{2}k{y}^2\\ E_1& =\frac{1}{2}k{Y}_1^2\\ E_2& =\frac{1}{2}k{Y}_2^2\\ ⟹\frac{E_1-E_2}{E_1}& =1-\frac{E_2}{E_1}\\ & =1-{\left(\frac{Y_2}{Y_1}\right)}^2\\ & =0.75⟹\text{75\% Energy Lost}\end{array}$$