Question$\_(15$ marks total$)$

a$)$ The one$-$story shear building shown in Figure $1\mathrm{.}1$ can be modelled in the first approximation as a SDOF system by regarding the columns as massless and the roof as rigid. Assume the columns are axially rigid and the mass $m$ can only translate horizontally, so that the columns undergo no rotation at the top.

i$)$ Formulate the equation of motion for the one$-$story building shown in Figure $1\mathrm{.}1$ and determine the natural frequency $($in $Hz).$ Formulate the equation of motion for the one$-$story building shown in Figure $1\mathrm{.}1.(3$ marks$)$

ii$)$ The properties of the shear building are $EI=22,500k\frac{N}{m},m=75$ton,$H=3m.$ determine the natural frequency $($in $Hz).(3$ marks$)$

ii$)$ If an additional column with a pinned supported at the ground is added to the centre of the shear building, what is the percentage increase in the natural frequency $($using the shear building properties in $($ii$)$ to do the calculation$)?(4$ marks$)$

b$)$ The damped spring$-$mass system shown in Figure $1\mathrm{.}2$ has a mass $m=150kg,$ a stiffness $k=6\mathrm{.}5{10}^6\frac{N}{m},$ and a damping cocfficient $c=5500N\frac{s}{m}.$ if the mass is subject to an initial displacement of $x(0)=25mm$ and suddenly released with zero initial velocity, determine the amplitude of free vibration of the mass after $4,8$ and $12$ oscillations. $(5$ marks$)$