$(a)$ \[ \nu^2=\frac{|a_1|}{|a_2|}\times\frac{|a_2|}{|a_3|}=e^{\varepsilon T_d}\] 両辺の対数をとると \[\ln\nu^2=\ln\frac{|a_1|}{|a_3|}=\varepsilon T_d\] \[2\ln\nu=2\delta=\ln\frac{|a_1|}{|a_3|}= \varepsilon T_d\] \[\delta=\frac{1}{2}\ln\frac{|a_1|}{|a_3|}=\frac{1}{2}\ln\frac{1}{0.0226}=1.894\\ \underline{\therefore \delta =1.89}\] $(b)$ \[\delta=\frac{\varepsilon T_d}{2}=\frac{\omega_n\cdot\zeta\cdot\frac{2\pi}{\omega_d}}{2}=\frac{\omega_n\cdot \zeta\cdot\pi}{\omega_n\sqrt{1-\zeta^2}}\] \begin{align}\delta^2(1-\zeta^2)&=(\pi\zeta)^2\\\delta^2-\delta^2\zeta^2&=(\pi\delta)^2\\\delta^2&=(\pi\zeta)^2+\delta^2\zeta^2\\&=(\pi^2+\delta^2)\zeta^2\end{align} \[ \begin{align} \zeta&=\sqrt{\frac{\delta^2}{\pi^2+\delta^2}}\\&=\sqrt{\frac{1.894^2}{\pi^2+1.894^2}}\\&=0.5163\\ \therefore\zeta&=\underline{0.516} \end{align} \] $(c)$ \begin{align}\omega_d&=\omega_n\sqrt{1-\zeta^2}\\&=\sqrt{\frac{k}{m}}\cdot\sqrt{1-\zeta^2}\\&=\sqrt{\frac{1200\,\rm{N/m}}{5\,\rm{kg}}}\cdot\sqrt{1-0.5163^2}\\&=13.26\,\rm{rad/s}\end{align}\[\underline{\therefore\omega_d=13.3\,\rm{rad/s}}\] \begin{align}f_d&=\frac{w_d}{2\pi}\\ &=\frac{13.26\,\rm{rad/s}}{2\pi}\\ &=2.110\,\rm{Hz}\end{align} \[\underline{\therefore f_d=2.11\,\rm{Hz}}\] $(d)$ \begin{align}c&=c_{cr}\cdot\zeta\\&=2\sqrt{mk}\cdot \zeta\\ &=2\sqrt{5\,\rm{kg}\times1200\,\rm{N/m}}\times0.5163\\ &=79.98\,\rm{N/(m/s)}\end{align} \[\underline{\therefore c=80.0\,\rm{N/(m/s)}}\]