$(15$ points$)$ Consider a fixed$-$free bar of a uniform cross$-$sectional area $A,$ a length $l,$ a modulus of elasticity $E,$ and a mass density $.$ Find the longitudinal vibration response of the bar subjected to the following initial conditions

$u(x,0)=U_0{x}^2,$ and $,\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}(x,0)=0$

$[$Ans: $[$:$u(x,t)=-\frac{16U_0{l}^2}{{}^3}{\displaystyle \underset{n=1}{\overset{}{}}},2,$dots$\frac{(-1{)}^n(2n-1)+2}{(2n-1{)}^3}sin(\frac{(2n-1)x}{2l})cos(\frac{(2n-1)ct}{2l})$