Q$3.$ For the given structural system in the figure, determine;

a$)(15$ p$.)$ natural frequencies and mode shapes

b$)(10$ p$.)$ damping coefficient $(c{?}_{(}))$ matrix by using Rayleigh damping coefficients. Use the first two modes to determine the $a_0$ and $a_1.$ Use $5\%$ damping for each mode.

c$)(10$ p$.)$ modal mass $($M$)$ and modal stiffness $($K$)$ matrices for the system,

d$)(5$ p$.)$ the displacement responses for undamped steady$-$state vibration, if $p_1(t)=2**p_2(t)=P_0**sin(2t).$

$E=200000$MPa;$I=347072m{m}^4$;$L=5m$;$m=10$ tons

Hint: Please be careful while constructing the stiffness matrix. Use only dynamic DOFs for the analysis. Make static condensation.