$[30\%]$ Given the single degree$-$of$-$freedom consisted of two identical beams connected by rigid massless link and supported by springs and hinge as depicted in Figure $1.$ The mass moment of inertia of each beam with respect to its center of gravity equals to

$J_G=\frac{m(2(L){)}^2}{12}$

Figure $1$ The beams are identical, connected by rigid massless link. Triangles represent hinge support fixed to the ground.

Please answer the following questions:

a$.[10\%]$ Draw the free body diagram of the system.

b$.[10\%]$ Derive the differential equation of motion of the system for small angular displacement.

c$.10\%$ Determine the natural frequency expression of the system as a function of $k,m,$ and $L.$

Useful formulas:

$m{x}^{}+c{x}^{}+kx=0$

${}_n=2f_n=\sqrt[\phantom{2}]{\frac{k}{m}},{}_d={}_n\sqrt[\phantom{2}]{1-{}^2}$

$=\frac{c}{c_c},c_c=2\sqrt[\phantom{2}]{km}$

$=\frac{2}{\sqrt[\phantom{2}]{1-{}^2}}=\frac{1}{N}ln(\frac{x_k}{x_{k+N}})$

$x(t)=A{e}^{-\{\begin{array}{c}{}_{n}t\end{array}}sin({}_{d}t+)$

$x(t)=C_1{e}^{(--\sqrt[\phantom{2}]{1-{}^2}){}_{n}t}+c_2{e}^{(-+\sqrt[\phantom{2}]{1-{}^2}){}_{n}t}$

$x(t)=(C_3+C_{4}t)e^{-{}_{n}t}$