$K E \ = \ \frac{1}{2} \ I_o \ \dot{θ}^2$

$= \frac{1}{2} [ \frac{mL^2}{12} \ + \ m \ (\frac{L}{2}^2] \theta^2$

$Ieq \ = \ \frac{mL^2}{12} \ + \ m \ (\frac{L}{2})^2$

$PE \ = \ \frac{1}{2} \ (K) \ (a \theta)^2$

$= \frac{1}{2} \ k.a^2 \ \theta^2$

$Keq \ = \ K.a^2 \ = \ 2 \times \ 10^5 \ \times \ 0.2^2 \ = \ 8000 \ N/m$

$W_n \ = \ \sqrt{ \frac{Keq}{Ieq}} \ = \ \sqrt{ \frac{8000}{0.726}}$

$W_n \ = \ 105 rad/s$

4] Work done = $- \int \ \ c.y \ dy$

Y = definition of damper = L $\theta$

$\dot{Y}$ = velocity of damper = L $\dot{θ}$

$ - \int \ C. L \theta d [L \theta]$

$ = - \int \ C. L^2 . \theta d \theta$

5] $r \ = \ \frac{w}{w_n} = \frac{100}{105}$ = 0.9523

6] $\theta_{st} \ = \ \frac{M_o}{Keq} = \frac{f_o \times L}{Keq}$

$= \frac{100 \times 1.1}{8000}$

= 0.01375 rad

7] $\xi \ = \ \frac{C_{eq}}{C_{el}} \ = \ \frac{CL^2}{2.I_{eq}. \ W_n} \ = \ \frac{ 30 \ \times \ (1.1)^2}{2 \ \times \ 0.726 \ \times \ 105}$

= 0.238

8] $\theta \ = \ \frac{0.01375}{\sqrt{ (1-0.01375^2)^2 \ + \ (2 \ \times 0.238 \ \times \ 0.01375)^2}}$

$\theta = 0.0297 \ rad$