$2.)$ Consider the system of two degrees$-$of$-$freedom given below with $m_1=3t,m_2=4t,k=240k\frac{N}{m}$ and ${}_n=5\%$;

a$.$ Write down the equations of motion of the system by including the ground motion $u_g(t)$ and evaluate the mass matrix m $,$ the rigidity matrix $k,$

b$.$ Determine the two circular frequencies and the periods of the free vibration ${}_1$ and $T_i.$

c$.$ Obtain the corresponding mode shapes $,$ and give their graphical representations,

d$.$ Check the orthogonality of the modes with respect to the mass matrix and the stiffness matrix ${}_1^{T}m{}_2$ and ${}_1^{T}k{}_2,$

e$.$ Evaluate the generalized masses and stiffnesses $M_i={}_i^{T}m{}_i$ and $K_i={}_i^{T}k{}_i$ and assess $w_i^2=\frac{K_i}{M_i},$

f$.$ Using the elastic design spectrum given to you in the first question $(S_{DS}=1\mathrm{.}041S_{D_1}=0\mathrm{.}3435),$ calculate the maximum displacement at the top story of the frame and the maximum base shear force of the system. $($Apply the SRSS rule$).$

Reminders:

$M_n=\frac{O}{{?}_n^T}[M]\frac{O}{{?}_n},L_n=\frac{O}{{?}_n^T}[M]\{l\},D_n=\frac{L_n}{M_n}\frac{S_{ae,n}}{{}_n^2},u_{n,\text{m}\text{a}\text{x}}=\frac{O}{{?}_n}{D}_n$

$\{u{\}}_{\text{m}\text{a}\text{x}}=\sqrt[\phantom{2}]{u_{1,\text{m}\text{a}\text{x}}^2+u_{2,\text{m}\text{a}\text{x}}^2}(SRSS$ Rule $o.)$