Do full written answer!! $3.$ As shown in the following figure, a two$-$story building is simplified as two masses $(2$ m and $\backslash ($ m $\backslash ))$ supported by six identical columns. Each column can be modeled as a linear spring $\backslash ($ k $\backslash ).$ Both $\backslash ($ m $\backslash )$ and $\backslash ($ k $\backslash )$ are given.

a$)$ Obtain the governing equations $(15$ points$)$

b$)$ Obtain the natural frequencies $(15$ points$)$

c$)$ Obtain the shapes of the natural modes $(10$ points$)$