$1.$ For the system given at the right hand side, determine:

a$)$ The stiffness matrix of the system,

b$)$ The dynamic stiffness, $($You can use any method to determine force$-$displacement relation at the given degree of freedom$)$

c$)$ The natural frequency of the system for mass $\backslash ($ m $\backslash ),$

d$)$ For an initial displacement and velocity of $0\mathrm{.}1$ m and $\backslash (2\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ at the end of the beam, determine the maximum support moment.

$2.$ For the system given at the right hand side with two columns and a rigid beam, assuming all members are axially rigid, determine

a$)$ the stiffness matrix in terms of EI and $\backslash (\backslash$alpha $\backslash ),$

b$)$ given $\backslash (\backslash$mathrm$\{$EI$\}=5\backslash$mathrm$\{$E$\}6\backslash$mathrm$\{$Nm$\}^\{2\},\backslash$mathrm$\{$~L$\}=3\backslash ),$ and $\backslash (\backslash$mathrm$\{$m$\}=500\backslash$mathrm$\{$~kg$\}\backslash ),$ determine the dynamic stiffness of the system in terms of $\backslash (\backslash$alpha $\backslash ).$

c$)$ the natural period in terms of $\backslash (\backslash$alpha $\backslash )$

d$)$ plot the natural period of the system for $\backslash (\backslash$alpha$=1\backslash$ldots $100\backslash ).$

e$)$ If an initial support displacement of $3$ cm is given to the base of the left hand beam, determine the equation for the lateral displacement of the beam.

f$)$ If an initial displacement of $5$ cm is given to the beam determine the equation of the rotation at the left support.